# Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 4 - Determinants and Matrices [Latest edition]

## Chapter 4: Determinants and Matrices

Exercise 4.1Exercise 4.2Exercise 4.3Miscellaneous Exercise 4(A)Exercise 4.4Exercise 4.5Exercise 4.6Exercise 4.7Miscellaneous Exercise 4(B)
Exercise 4.1 [Pages 63 - 64]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Exercise 4.1 [Pages 63 - 64]

Exercise 4.1 | Q 1. (i) | Page 63

Find the value of determinant :

|(2, -4),(7, -15)|

Exercise 4.1 | Q 1. (ii) | Page 63

Find the value of determinant :

|(2"i", 3),(4, -"i")|

Exercise 4.1 | Q 1. (iii) | Page 63

Find the value of determinant :

|(3, -4, 5),(1, 1, -2),(2, 3, 1)|

Exercise 4.1 | Q 1. (iv) | Page 63

Find the value of determinant :

|("a", "h", "g"),("h", "b", "f"),("g", "f", "c")|

Exercise 4.1 | Q 2. (i) | Page 63

Find the value of x if

|(x^2 - x + 1, x + 1),(x + 1, x + 1)| = 0

Exercise 4.1 | Q 2. (ii) | Page 63

Find the value of x if

|(x, -1, 2),(2x, 1, -3),(3, -4, 5)| = 29

Exercise 4.1 | Q 3 | Page 63

Find x and y if |(4"i", "i"^3, 2"i"),(1, 3"i"^2, 4),(5, -3, "i")| = x + iy where i2 = – 1

Exercise 4.1 | Q 4 | Page 63

Find the minor and cofactor of element of the determinant

D = |(2, -1, 3),(1, 2, -1),(5, 7, 2)|

Exercise 4.1 | Q 5. (a) | Page 64

Evaluate A = |(2, -3,5),(6, 0, 4),(1, 5, -7)| Also find minor and cofactor of elements in the 2nd row of determinant and verify − a21.M21 + a22.M22 − a23.M23 = value of A

where M21, M22 , M23 are minor of a21 , a22, a23  and C21, C22, C23 are cofactor of a21, a22, a23

Exercise 4.1 | Q 5. (b) | Page 64

Evaluate A = |(2, -3,5),(6, 0, 4),(1, 5, -7)| Also find minor and cofactor of elements in the 2nd row of determinant and verify a21 C21 + a22 C22 + a23 C23 = value of A

where M21, M22, M23 are minors of a21, a22, a23 and

C21, C22, C23 are cofactors of a21, a22, a23.

Exercise 4.1 | Q 6 | Page 64

Find the value of determinant expanding along third column

|(-1, 1, 2),(-2, 3, -4),(-3, 4, 0)|

Exercise 4.2 [Pages 67 - 68]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Exercise 4.2 [Pages 67 - 68]

Exercise 4.2 | Q 1. (i) | Page 67

Without expanding evaluate the following determinant:

|(1, "a", "b" + "c"),(1, "b", "c" + "a"),(1, "c", "a" + "b")|

Exercise 4.2 | Q 1. (ii) | Page 67

Without expanding evaluate the following determinant:

|(2, 3, 4),(5, 6, 8),(6x, 9x, 12x)|

Exercise 4.2 | Q 1. (iii) | Page 67

Without expanding evaluate the following determinant:

|(2, 7, 65),(3, 8, 75),(5, 9, 86)|

Exercise 4.2 | Q 2 | Page 68

Prove that |(x + y, y + z, z + x),(z + x, x + y, y + x),(y + z, z + x, x + y)| = 2|(x, y, z),(z, x, y),(y, z, x)|

Exercise 4.2 | Q 3. (i) | Page 68

Using properties of determinant show that

|("a" + "b", "a", "b"),("a", "a" + "c", "c"),("b", "c", "b" + "c")| = 4abc

Exercise 4.2 | Q 3. (ii) | Page 68

Using properties of determinant show that

|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)| = 0

Exercise 4.2 | Q 4. (i) | Page 68

Solve the following equation:

|(x + 2, x + 6, x - 1),(x + 6, x - 1, x + 2),(x - 1, x + 2, x + 6)| = 0

Exercise 4.2 | Q 4. (ii) | Page 68

Solve the following equation:

|(x -1, x, x - 2),(0, x - 2, x - 3),(0, 0, x - 3)| = 0

Exercise 4.2 | Q 5 | Page 68

If  |(4 + x, 4 - x, 4 - x),(4 - x,4 + x,4 - x),(4 - x,4 - x, 4 + x)| = 0, then find the values of x.

Exercise 4.2 | Q 6 | Page 68

Without expanding determinants show that

|(1, 3, 6),(6, 1, 4),(3, 7, 12)| + 4|(2, 3, 3),(2, 1, 2),(1, 7, 6)| = 10|(1, 2, 1),(3, 1, 7),(3, 2, 6)|

Exercise 4.3 [Pages 74 - 75]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Exercise 4.3 [Pages 74 - 75]

Exercise 4.3 | Q 1. (i) | Page 74

Solve the following linear equations by using Cramer’s Rule:

x + y + z  = 6, x – y + z = 2, x + 2y – z = 2

Exercise 4.3 | Q 1. (ii) | Page 74

Solve the following linear equations by using Cramer’s Rule:

x + y − 2z = –10, 2x + y – 3z = –1, 4x + 6y + z = 2

Exercise 4.3 | Q 1. (iii) | Page 74

Solve the following linear equations by using Cramer’s Rule:

x + z = 1, y + z = 1, x + y = 4

Exercise 4.3 | Q 1. (iv) | Page 75

Solve the following linear equations by using Cramer’s Rule:

(-2)/x - 1/y - 3/x = 3, 2/x - 3/y + 1/z = -13 and 2/x - 3/z = – 11

Exercise 4.3 | Q 2 | Page 75

The sum of three numbers is 15. If the second number is subtracted from the sum of first and third numbers then we get 5. When the third number is subtracted from the sum of twice the first number and the second number, we get 4. Find the three numbers.

Exercise 4.3 | Q 3. (i) | Page 75

Examine the consistency of the following equation:

2x − y + 3 = 0, 3x + y − 2 = 0, 11x + 2y − 3 = 0

Exercise 4.3 | Q 3. (ii) | Page 75

Examine the consistency of the following equation:

2x + 3y − 4 = 0, x + 2y = 3, 3x + 4y + 5 = 0

Exercise 4.3 | Q 3. (iii) | Page 75

Examine the consistency of the following equation:

x + 2y −3 = 0, 7x + 4y − 11 = 0, 2x + 4y − 6 = 0

Exercise 4.3 | Q 4. (i) | Page 75

Find k if the following equations are consistent:

2x + 3y - 2 = 0, 2x + 4y − k = 0, x − 2y + 3k =0

Exercise 4.3 | Q 4. (ii) | Page 75

Find k if the following equations are consistent:

kx +3y + 1 = 0, x + 2y + 1 = 0, x + y = 0

Exercise 4.3 | Q 5. (i) | Page 75

Find the area of triangle whose vertices are

A(5, 8), B(5, 0) C(1, 0)

Exercise 4.3 | Q 5. (ii) | Page 75

Find the area of triangle whose vertices are

"P"(3/2, 1), "Q"(4, 2), "R"(4, (-1)/2)

Exercise 4.3 | Q 5. (iii) | Page 75

Find the area of triangle whose vertices are

M(0, 5), N(−2, 3), T(1, −4)

Exercise 4.3 | Q 6 | Page 75

Find the area of quadrilateral whose vertices are

A(−3, 1), B(−2, −2), C(1, 4), D(3, −1)

Exercise 4.3 | Q 7 | Page 75

Find the value of k, if the area of triangle whose vertices are P(k, 0), Q(2, 2), R(4, 3) is 3/2 "sq.unit"

Exercise 4.3 | Q 8. (i) | Page 75

Examine the collinearity of the following set of point:

A(3, −1), B(0, −3), C(12, 5)

Exercise 4.3 | Q 8. (ii) | Page 75

Examine the collinearity of the following set of point:

P(3, −5), Q(6, 1), R(4, 2)

Exercise 4.3 | Q 8. (iii) | Page 75

Examine the collinearity of the following set of point:

"L"(0, 1/2), "M"(2, -1), "N"(-4, 7/2)

Miscellaneous Exercise 4(A) [Pages 75 - 76]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Miscellaneous Exercise 4(A) [Pages 75 - 76]

Miscellaneous Exercise 4(A) | Q I. (1) | Page 75

Select the correct option from the given alternatives:

The determinant D = |("a", "b", "a" + "b"),("b", "c", "b" + "c"),("a" + "b", "b" + "c", 0)| = 0 if

• a, b, c are in A.P.

• a, b, c are in G.P

• a, b, c are in H.P.

• α is root of ax2 + 2bx + c = 0

Miscellaneous Exercise 4(A) | Q I. (2) | Page 75

Select the correct option from the given alternatives:

If |("x"^"k", "x"^("k" + 2), "x"^("k" + 3)),("y"^"k", "y"^("k" + 2), "y"^("k" + 3)),("z"^"k", "z"^("k" + 2), "z"^("k" + 3))| = (x - y) (y - z) (z - x)(1/"x"+ 1/"y" + 1/"z")  then

• k= –3

• k = –1

• k = 1

• k = 3

Miscellaneous Exercise 4(A) | Q I. (3) | Page 75

Select the correct option from the given alternatives:

Let D = |(sintheta*cosphi, sintheta*sinphi, costheta),(costheta*cosphi, costheta*sinphi, -sintheta),(-sintheta*sinphi, sintheta*cosphi, 0)| then

• D is independent of θ

• D is independent of Φ

• D is a constant

• "dD"/"d" at theta = pi/2 is equal to 0

Miscellaneous Exercise 4(A) | Q I. (4) | Page 75

Select the correct option from the given alternatives:

The value of a for which system of equation a3x + (a + 1)3 y + (a + 2)3z = 0 ax + (a +1)y + (a + 2)z = 0 and x + y + z = 0 has non zero Soln. is

• 0

• –1

• 1

• 2

Miscellaneous Exercise 4(A) | Q I. (5) | Page 76

Select the correct option from the given alternatives:

|("b" + "c", "c" + "a", "a" + "b"),("q" + "r", "r" + "p", "p" + "q"),(y + z, z + x, x + y)| =

• 2|("c", "b", "a"),("r", "q", "p"),(z, y, x)|

• 2|("b", "a", "c"),("q", "p", "r"),(y, x, z)|

• 2|("a", "b", "c"),("p", "q", "r"),(x, y, z)|

• 2|("a", "c", "b"),("p", "r", "q"),(x, z, y)|

Miscellaneous Exercise 4(A) | Q I. (6) | Page 76

Select the correct option from the given alternatives:

The system 3x – y + 4z = 3, x + 2y –3z = –2 and 6x + 5y + λz = –3 has at least one Solution when

• λ = –5

• λ = 5

• λ = 3

• λ = –13

Miscellaneous Exercise 4(A) | Q I. (7) | Page 76

Select the correct option from the given alternatives:

If x = –9 is a root of |(x, 3, 7),(2, x, 2),(7, 6, x)| = 0 has other two roots are

• 2, –7

• –2, 7

• 2, 7

• 2, –7

Miscellaneous Exercise 4(A) | Q I. (8) | Page 76

Select the correct option from the given alternatives:

If |(6"i", -3"i", 1),(4, 3"i", -1),(20, 3, "i")| = x + iy then

• x = 3 , y = 1

• x = 1 , y = 3

• x = 0 , y = 3

• x = 0 , y = 0

Miscellaneous Exercise 4(A) | Q I. (9) | Page 76

Select the correct option from the given alternatives:

If A(0,0), B(1,3) and C(k,0) are vertices of triangle ABC whose area is 3 sq.units then value of k is

• 2

• –3

• 3 or −3

• –2 or +2

Miscellaneous Exercise 4(A) | Q I. (10) | Page 76

Select the correct option from the given alternatives:

Which of the following is correct

• Determinant is square matrix

• Determinant is number associated to matrix

• Determinant is number associated to square matrix

• None of these

Miscellaneous Exercise 4(A) [Pages 76 - 77]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Miscellaneous Exercise 4(A) [Pages 76 - 77]

Miscellaneous Exercise 4(A) | Q II. (1) (i) | Page 76

Evaluate |(2, -5, 7),(5, 2, 1),(9, 0, 2)|

Miscellaneous Exercise 4(A) | Q II. (1) (ii) | Page 76

Evaluate |(1, -3, 12),(0, 2, -4),(9, 7, 2)|

Miscellaneous Exercise 4(A) | Q II. (2) | Page 76

Evaluate determinant along second column

|(1, -1, 2),(3, 2, -2),(0, 1, -2)|

Miscellaneous Exercise 4(A) | Q II. (3) (i) | Page 76

Evaluate |(2, 3, 5),(400, 600, 1000),(48, 47, 18)| by using properties

Miscellaneous Exercise 4(A) | Q II. (3) (ii) | Page 76

Evaluate |(101, 102, 103),(106, 107, 108),(1, 2, 3)| by using properties

Miscellaneous Exercise 4(A) | Q II. (4) (i) | Page 76

Find minor and cofactor of elements of the determinant:

|(-1, 0, 4),(-2, 1, 3),(0, -4, 2)|

Miscellaneous Exercise 4(A) | Q II. (4) (ii) | Page 76

Find minor and cofactor of elements of the determinant:

|(1, -1, 2),(3, 0, -2),(1, 0, 3)|

Miscellaneous Exercise 4(A) | Q II. (5) (i) | Page 76

Find the value of x if

|(1, 4, 20),(1, -2, -5),(1, 2x, 5x^2)| = 0

Miscellaneous Exercise 4(A) | Q II. (5) (ii) | Page 76

Find the value of x if

|(1, 2x, 4x),(1, 4, 16),(1, 1, 1)| = 0

Miscellaneous Exercise 4(A) | Q II. (6) | Page 76

By using properties of determinant prove that |(x + y, y + z, z + x),(z, x, y),(1, 1, 1)| = 0

Miscellaneous Exercise 4(A) | Q II. (7) (i) | Page 77

Without expanding determinant show that

|("b" + "c", "bc", "b"^2"c"^2),("c" + "a", "ca", "c"^2"a"^2),("a" + "b", "ab", "a"^2"b"^2)| = 0

Miscellaneous Exercise 4(A) | Q II. (7) (ii) | Page 77

Without expanding determinant show that

|(x"a", y"b", z"c"),("a"^2, "b"^2, "c"^2),(1, 1, 1)| = |(x, y, z),("a", "b", "c"),("bc", "ca", "ab")|

Miscellaneous Exercise 4(A) | Q II. (7) (iii) | Page 77

Without expanding determinant show that

|(l, "m", "n"),("e", "d", "f"),("u", "v", "w")| = |("n", "f", "w"),(l, "e", "w"),("m", "d", "v")|

Miscellaneous Exercise 4(A) | Q II. (7) (iv) | Page 77

Without expanding determinant show that

|(0, "a", "b"),(-"a", 0, "c"),(-"b", -"c", 0)| = 0

Miscellaneous Exercise 4(A) | Q II. (8) | Page 77

If |("a", 1, 1),(1, "b", 1),(1, 1, "c")| = 0 then show that 1/(1 - "a") + 1/(1 - "b") + 1/(1 - "c") = 1

Miscellaneous Exercise 4(A) | Q II. (9) (i) | Page 77

Solve the following linear equations by Cramer’s Rule:

2x − y + z = 1, x + 2y + 3z = 8, 3x + y − 4z =1

Miscellaneous Exercise 4(A) | Q II. (9) (ii) | Page 77

Solve the following linear equations by Cramer’s Rule:

1/x + 1/y = 3/2, 1/y + 1/z = 5/6, 1/z + 1/x = 4/3

Miscellaneous Exercise 4(A) | Q II. (9) (iii) | Page 77

Solve the following linear equations by Cramer’s Rule:

2x+ 3y + 3z = 5 , x − 2y + z = – 4 , 3x – y – 2z = 3

Miscellaneous Exercise 4(A) | Q II. (9) (iv) | Page 77

Solve the following linear equations by Cramer’s Rule:

x – y + 2z = 7 , 3x + 4y – 5z = 5 , 2x – y + 3z = 12

Miscellaneous Exercise 4(A) | Q II. (10) (i) | Page 77

Find the value of k, if the following equations are consistent:

(k + 1)x + (k – 1)y + (k – 1) = 0
(k – 1)x + (k + 1)y + (k – 1) = 0
(k – 1)x + (k – 1)y + (k + 1) = 0

Miscellaneous Exercise 4(A) | Q II. (10) (ii) | Page 77

Find the value of k, if the following equations are consistent:

3x + y − 2 = 0 kx + 2y − 3 = 0 and 2x − y = 3

Miscellaneous Exercise 4(A) | Q II. (10) (iii) | Page 77

Find the value of k if the following equation are consistent:

(k − 2)x +(k − 1)y =17 , (k − 1)x + (k − 2)y = 18 and x + y = 5

Miscellaneous Exercise 4(A) | Q II. (11) (i) | Page 77

Find the area of triangle whose vertices are A(−1, 2), B(2, 4), C(0, 0)

Miscellaneous Exercise 4(A) | Q II. (11) (ii) | Page 77

Find the area of triangle whose vertices are P(3, 6), Q(−1, 3), R(2, −1)

Miscellaneous Exercise 4(A) | Q II. (11) (iii) | Page 77

Find the area of triangle whose vertices are L(1, 1), M(−2, 2), N(5, 4)

Miscellaneous Exercise 4(A) | Q II. (12) (i) | Page 77

Find the value of k:

If area of triangle is 4 square unit and vertices are P(k, 0), Q(4, 0), R(0, 2)

Miscellaneous Exercise 4(A) | Q II. (12) (ii) | Page 77

Find the value of k:

If area of triangle is 33/2 square unit and vertices are L(3, −5), M(−2, k), N(1, 4)

Miscellaneous Exercise 4(A) | Q II. (13) | Page 77

Find the area of quadrilateral whose vertices are A(0, −4), B(4, 0), C(−4, 0), D(0, 4)

Miscellaneous Exercise 4(A) | Q II. (14) | Page 77

An amount of ₹ 5000 is put into three investments at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is ₹ 350. If the combined income from the first two investments is ₹ 7000 more than the income from the third. Find the amount of each investment.

Miscellaneous Exercise 4(A) | Q II. (15) | Page 77

Show that the lines x − y = 6, 4x − 3y = 20 and 6x + 5y + 8 = 0 are concurrent. Also find the point of concurrence

Miscellaneous Exercise 4(A) | Q II. (16) (i) | Page 77

Show that the following points are collinear by determinant:

L(2,5), M(5,7), N(8,9)

Miscellaneous Exercise 4(A) | Q II. (16) (ii) | Page 77

Show that the following points are collinear by determinant:

P(5,1), Q(1,−1), R(11,4)

Exercise 4.4 [Pages 82 - 83]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Exercise 4.4 [Pages 82 - 83]

Exercise 4.4 | Q 1. (i) | Page 82

Construct a matrix A = = [aij]3×2 whose element aij is given by

aij = (("i" - "j")^2)/(5 - "i")

Exercise 4.4 | Q 1. (ii) | Page 82

Construct a matrix A = = [aij]3×2 whose element aij is given by

aij = i – 3j

Exercise 4.4 | Q 1. (iii) | Page 82

Construct a matrix A = = [aij]3×2 whose element aij is given by

aij = (("i" + "j")^3)/5

Exercise 4.4 | Q 2. (i) | Page 82

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

[(3, -2, 4),(0, 0, -5),(0, 0, 0)]

Exercise 4.4 | Q 2. (ii) | Page 82

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

[(0, 4, 7),(-4, 0, -3),(-7, 3, 0)]

Exercise 4.4 | Q 2. (iii) | Page 82

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

[(5),(4),(-3)]

Exercise 4.4 | Q 2. (iv) | Page 82

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

[9   sqrt(2)  -3]

Exercise 4.4 | Q 2. (v) | Page 83

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

[(6, 0),(0, 6)]

Exercise 4.4 | Q 2. (vi) | Page 83

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

[(2, 0, 0),(3, -1, 0),(-7, 3, 1)]

Exercise 4.4 | Q 2. (vii) | Page 83

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

[(3, 0, 0),(0, 5, 0),(0, 0, 1/3)]

Exercise 4.4 | Q 2. (viii) | Page 83

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

[(10, -15, 27),(-15, 0, sqrt(34)),(27, sqrt(34), 5/3)]

Exercise 4.4 | Q 2. (ix) | Page 83

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

[(1, 0, 0),(0, 1, 0),(0, 0, 1)]

Exercise 4.4 | Q 2. (x) | Page 83

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

[(0, 0, 1),(0, 1, 0),(1, 0, 0)]

Exercise 4.4 | Q 3. (i) | Page 83

Identify the following matrix is singular or non-singular?

[("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")]

Exercise 4.4 | Q 3. (ii) | Page 83

Identify the following matrix is singular or non-singular?

[(5, 0, 5),(1, 99, 100),(6, 99, 105)]

Exercise 4.4 | Q 3. (iii) | Page 83

Identify the following matrix is singular or non-singular?

[(3, 5, 7),(-2, 1, 4),(3, 2, 5)]

Exercise 4.4 | Q 3. (iv) | Page 83

identify the following matrix is singular or non-singular?

[(7, 5),(-4, 7)]

Exercise 4.4 | Q 4. (i) | Page 83

Find k if the following matrix is singular:

[(7, 3),(-2, "k")]

Exercise 4.4 | Q 4. (ii) | Page 83

Find k if the following matrix is singular:

[(4, 3, 1),(7, "k", 1),(10, 9, 1)]

Exercise 4.4 | Q 4. (iii) | Page 83

Find k if the following matrix is singular:

[("k" - 1, 2, 3),(3, 1, 2),(1, -2, 4)]

Exercise 4.4 | Q 5 | Page 83

If A = [(5, 1, -1),(3, 2, 0)], Find (AT)T.

Exercise 4.4 | Q 6 | Page 83

If A = [(7, 3, 1),(-2, -4, 1),(5, 9, 1)], Find (AT)T.

Exercise 4.4 | Q 7 | Page 83

Find a, b, c if [(1, 3/5, "a"),("b", -5, -7),(-4, "c", 0)] is a symmetric matrix.

Exercise 4.4 | Q 8 | Page 83

Find x, y, z If [(0, -5"i", x),(y, 0, z),(3/2, -sqrt(2), 0)] is a skew symmetric matrix.

Exercise 4.4 | Q 9. (i) | Page 83

The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:

[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]

Exercise 4.4 | Q 9. (ii) | Page 83

The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:

[(2, 5, 1),(-5, 4, 6),(-1, -6, 3)]

Exercise 4.4 | Q 9. (iii) | Page 83

The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:

[(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)]

Exercise 4.4 | Q 10 | Page 83

Construct the matrix A = [aij]3×3 where aij = i − j. State whether A is symmetric or skew-symmetric.

Exercise 4.5 [Pages 86 - 87]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Exercise 4.5 [Pages 86 - 87]

Exercise 4.5 | Q 1. (i) | Page 86

If A = [(2, -3),(5, -4),(-6, 1)], "B" = [(-1, 2),(2, 2), (0, 3)] and "C" = [(4,  3),(-1, 4),(-2, 1)] Show that A + B = B + A

Exercise 4.5 | Q 1. (ii) | Page 86

If A = [(2, -3),(5, -4),(-6, 1)], "B" = [(-1, 2),(2, 2), (0, 3)] and "C" = [(4,  3),(-1, 4),(-2, 1)] Show that (A + B) + C = A + (B + C)

Exercise 4.5 | Q 2 | Page 87

If A = [(1, -2),(5, 3)], "B" = [(1, -3),(4, -7)], then find the matrix A – 2B + 6I, where I is the unit matrix of order 2.

Exercise 4.5 | Q 3 | Page 87

If A = [(1, 2, -3),(-3, 7, -8),(0, -6, 1)], "B" = [(9, -1, 2),(-4, 2, 5),(4, 0, -3)] then find the matrix C such that A + B + C is a zero matrix

Exercise 4.5 | Q 4 | Page 87

If A = [(1, -2),(3, -5),(-6, 0)], "B" = [(-1, -2),(4, 2),(1, 5)] and "C" = [(2, 4),(-1, -4),(-3, 6)], find the matrix X such that 3A – 4B + 5X = C.

Exercise 4.5 | Q 5 | Page 87

Solve the following equations for X and Y, if

3X – Y = [(1, -1),(-1, 1)] and "X" - 3"Y" = [(0, -1),(0, -1)]

Exercise 4.5 | Q 6 | Page 87

Find matrices A and B, if 2"A" - "B" = [(6, -6, 0),(-4, 2, 1)] and "A" - 2"B" = [(3, 2, 8),(-2, 1, -7)]

Exercise 4.5 | Q 7 | Page 87

Simplify, costheta[(costheta, sintheta),(-sintheta, costheta)] + sintheta[(sintheta, -costheta),(costheta, sintheta)]

Exercise 4.5 | Q 8 | Page 87

If A = [("i", 2"i"),(-3, 2)] and "B" = [(2"i", "i"),(2, -3)], where sqrt(-1) = i,, find A + B and A – B. Show that A + B is a singular. Is A – B a singular ? Justify your answer.

Exercise 4.5 | Q 9 | Page 87

Find x and y, if [(2x + y, -1, 1),(3, 4y, 4)] + [(-1, 6, 4),(3, 0, 3)] = [(3, 5, 5),(6, 18, 7)]

Exercise 4.5 | Q 10 | Page 87

If = [(2"a" + "b", 3"a" - "b"),("c" + 2"d", 2"c" - "d")] = [(2, 3),(4, -1)], find a, b, c and d.

Exercise 4.5 | Q 11. (i) | Page 87

There are two book shops owned by Suresh and Ganesh. Their sales (in Rupees) for books in three subject – Physics, Chemistry and Mathematics for two months, July and August 2017 are given by two matrices A and B.

July sales (in Rupees), Physics Chemistry Mathematics.

A = [(5600, 6750, 8500),(6650, 7055, 8905)]"First Row Suresh"/"Second Row Ganesh"

August sales(in Rupees), Physics Chemistry Mathematics

B = [(6650, 7055, 8905),(7000, 7500, 10200)]"First Row Suresh"/"Second Row Ganesh" then,

Find the increase in sales in Rupees from July to August 2017.

Exercise 4.5 | Q 11. (ii) | Page 87

There are two book shops owned by Suresh and Ganesh. Their sales (in Rupees) for books in three subject – Physics, Chemistry and Mathematics for two months, July and August 2017 are given by two matrices A and B.

July sales (in Rupees), Physics Chemistry Mathematics.

A = [(5600, 6750, 8500),(6650, 7055, 8905)]"First Row Suresh"/"Second Row Ganesh"

August sales(in Rupees), Physics Chemistry Mathematics

B = [(6650, 7055, 8905),(7000, 7500, 10200)]"First Row Suresh"/"Second Row Ganesh" then,

If both book shops got 10 % profit in the month of August 2017, find the profit for each book seller in each subject in that month

Exercise 4.6 [Pages 94 - 95]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Exercise 4.6 [Pages 94 - 95]

Exercise 4.6 | Q 1. (i) | Page 94

Evaluate: [(3),(2),(1)][(2,-4,3)]

Exercise 4.6 | Q 1. (ii) | Page 94

Evaluate : [2  -1   3][(4),(3),(1)]

Exercise 4.6 | Q 2 | Page 94

If A = [(1, -3),(4, 2)], "B" = [(4, 1),(3, -2)] show that AB ≠ BA.

Exercise 4.6 | Q 3 | Page 94

If A = [(-1, 1, 1),(2, 3, 0),(1, -3, 1)], "B" = [(2, 1, 4),(3, 0, 2),(1, 2, 1)]. State whether AB = BA? Justify your answer.

Exercise 4.6 | Q 4. (i) | Page 94

Show that AB = BA where,

A = [(-2, 3, -1),(-1, 2, 1),(-6, 9, -4)], "B" = [(1, 3, -1),(2, 2, -1),(3, 0, -1)]

Exercise 4.6 | Q 4. (ii) | Page 94

Show that AB=BA where,

A = [(costheta, sintheta),(sintheta, costheta)], "B"[(costheta, -sintheta),(sintheta, costheta)]

Exercise 4.6 | Q 5 | Page 94

If A = [(4, 8),(-2, -4)], prove that A2 = 0

Exercise 4.6 | Q 6. (i) | Page 94

Verify A(BC) = (AB)C of the following case:

A = [(1, 0, 1),(2, 3, 0),(0, 4, 5)], "B" = [(2, -2),(-1, 1),(0, 3)] and "C" = [(3, 2, -1),(2, 0, -2)]

Exercise 4.6 | Q 6. (ii) | Page 94

Verify A(BC) = (AB)C of the following case:

A = [(2, 4, 3),(-1, 3, 2)], "B" = [(2, -2),(3, 3),(-1, 1)], "C" = [(3, 1),(1, 3)]

Exercise 4.6 | Q 7. (i) | Page 94

Verify that A(B + C) = AB + BC of the following matrix:

A = [(4, -2),(2, 3)], "B" = [(-1, 1),(3, -2)] and "C" = [(4, 1),(2, -1)]

Exercise 4.6 | Q 7. (ii) | Page 94

Verify that A(B + C) = AB + BC of the following matrix:

A = [(1, -1, 3),(2, 3, 2)], "B" = [(1, 0),(-2, 3),(4, 3)], "C" = [(1, 2),(-2, 0),(4, -3)]

Exercise 4.6 | Q 8 | Page 94

If A = [(1, -2),(5, 6)], "B" = [(3, -1),(3, 7)], Find AB-2I, where I is unit matrix of order 2.

Exercise 4.6 | Q 9 | Page 94

If A = [(4, 3, 2),(-1, 2, 0)], "B" = [(1, 2),(-1, 0),(1, -2)] show that matrix AB is non singular

Exercise 4.6 | Q 10 | Page 94

If A = [(1, 2, 0),(5, 4, 2),(0, 7, -3)], find the product (A + I)(A − I)

Exercise 4.6 | Q 11 | Page 94

A = [(alpha, 0),(1, 1)], "B" = [(1, 0),(2, 1)] find α, if A2 = B.

Exercise 4.6 | Q 12 | Page 95

If A = [(1, 2, 2),(2, 1, 2),(2, 2, 1)], Show that A2 – 4A is a scalar matrix

Exercise 4.6 | Q 13 | Page 95

If A = [(1, 0),(-1, 7)], find k so that A2 – 8A – kI = O, where I is a unit matrix and O is a null matrix of order 2.

Exercise 4.6 | Q 14 | Page 95

If A = [(8, 4),(10, 5)], "B" = [(5, -4),(10, -8)] show that (A + B)2 = A2 + AB + B2

Exercise 4.6 | Q 15 | Page 95

If A = [(3, 1),(-1, 2)], prove that A2 – 5A + 7I = 0, where I is unit matrix of order 2

Exercise 4.6 | Q 16 | Page 95

If A = [(3, 4),(-4, 3)] and "B" = [(2, 1),(-1, 2)], show that (A + B)(A – B) = A2 – B

Exercise 4.6 | Q 17 | Page 95

If A = [(1, 2),(-1, -2)], "B" = [(2, "a"),(-1, "b")] and if (A + B)2 = A2 – B2 . find values of a and b

Exercise 4.6 | Q 18 | Page 95

Find matrix X such that AX = B, where A = [(1, -2),(-2, 1)] and B = [(-3),(-1)]

Exercise 4.6 | Q 19 | Page 95

Find k, if A= [(3, -2),(4, -2)] and if A2 = kA – 2I

Exercise 4.6 | Q 20 | Page 95

Find x, if [(1, "x", 1)][(1, 2, 3),(4, 5, 6),(3, 2, 5)] [(1),(-2), (3)] = 0

Exercise 4.6 | Q 21 | Page 95

Find x and y, if {4[(2, -1, 3),(1, 0, 2)] -[(3, -3, 4),(2, 1, 1)]} [(2),(-1),(1)] = [(x),(y)]

Exercise 4.6 | Q 22 | Page 95

Find x, y, z if {3[(2, 0),(0, 2),(2, 2)] -4[(1, 1),(-1, 2),(3, 1)]} [(1),(2)] = [("x" - 3),("y" - 1),(2"z")]

Exercise 4.6 | Q 23 | Page 95

If A = [(cosalpha, sinalpha),(-sinalpha, cosalpha)], show that "A"^2=[(cos2alpha, sin2alpha),(-sin2alpha, cos2alpha)]

Exercise 4.6 | Q 24 | Page 95

If A = [(1, 2),(3, 5)] B = [(0, 4),(2, -1)], show that AB ≠ BA, but |AB| = IAl·IBI

Exercise 4.6 | Q 25 | Page 95

Jay and Ram are two friends in a class. Jay wanted to buy 4 pens and 8 notebooks, Ram wanted to buy 5 pens and 12 notebooks. Both of them went to a shop. The price of a pen and a notebook which they have selected was Rs.6 and Rs.10. Using Matrix multiplication, find the amount required from each one of them

Exercise 4.7 [Pages 97 - 98]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Exercise 4.7 [Pages 97 - 98]

Exercise 4.7 | Q 1. (i) | Page 97

Find AT, if A = [(1, 3),(-4, 5)]

Exercise 4.7 | Q 1. (ii) | Page 97

Find AT, if A = [(2, -6, 1),(-4, 0, 5)]

Exercise 4.7 | Q 2 | Page 97

If [aij]3x3 where aij = 2(i – j). Find A and AT. State whether A and AT are symmetric or skew-symmetric matrices?

Exercise 4.7 | Q 3 | Page 97

If A = [(5, -3),(4, -3),(-2, 1)], Prove that (2A)T = 2AT

Exercise 4.7 | Q 4 | Page 97

If A = [(1, 2, -5),(2, -3, 4),(-5, 4, 9)], Prove that (3A)T = 3AT

Exercise 4.7 | Q 5 | Page 97

If A = [(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)] where i = sqrt(-1) Prove that AT = – A

Exercise 4.7 | Q 6. (i) | Page 97

If A = [(2, -3),(5, -4),(-6, 1)], B = [(2, 1),(4, -1),(-3, 3)] and C = [(1, 2),(-1, 4),(-2, 3)] then show that (A + B) = AT + BT

Exercise 4.7 | Q 6. (ii) | Page 97

If A = [(2, -3),(5, -4),(-6, 1)], B = [(2, 1),(4, -1),(-3, 3)] and C = [(1, 2),(-1, 4),(-2, 3)] then show that (A – C)T = AT – CT

Exercise 4.7 | Q 7 | Page 97

If A = [(5, 4),(-2, 3)] and B = [(-1, 3),(4, -1)], then find CT , such that 3A – 2B + C = I, where I is the unit matrix of order 2

Exercise 4.7 | Q 8. (i) | Page 98

If A [(7, 3, 0),(0, 4, -2)], B = [(0, -2, 3),(2, 1, -4)] then find AT + 4BT

Exercise 4.7 | Q 8. (ii) | Page 98

If A [(7, 3, 0),(0, 4, -2)], B = [(0, -2, 3),(2, 1, -4)] then find 5AT – 5BT

Exercise 4.7 | Q 9 | Page 98

If A = [(1, 0, 1),(3, 1, 2)], B = [(2, 1, -4),(3, 5, -2)] and C = [(0, 2, 3),(-1, 1, 0)], verify that (A + 2B + 2C)T = AT + 2BT + 3CT

Exercise 4.7 | Q 10 | Page 98

If A = [(-1, 2, 1),(-3, 2, -3)] and B = [(2, 1),(-3, 2),(-1, 3)], prove that (A + BT)T = AT + B

Exercise 4.7 | Q 11. (i) | Page 98

Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where

A = [(1, 2, 4),(3, 2, 1),(-2, -3, 2)]

Exercise 4.7 | Q 11. (ii) | Page 98

Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where

A = [(5, 2, -4),(3, -7, 2),(4, -5, -3)]

Exercise 4.7 | Q 12. (i) | Page 98

Express the following matrix as the sum of a symmetric and a skew symmetric matrix

[(4, -2),(3, -5)]

Exercise 4.7 | Q 12. (ii) | Page 98

Express the following matrix as the sum of a symmetric and a skew symmetric matrix

[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]

Exercise 4.7 | Q 13. (i) | Page 98

If A = [(2, -1),(3, -2),(4, 1)] and B = [(0, 3, -4),(2, -1, 1)], verify that  (AB)T = BT AT

Exercise 4.7 | Q 13. (ii) | Page 98

If A = [(2, -1),(3, -2),(4, 1)] and B = [(0, 3, -4),(2, -1, 1)], verify that (BA)T = AT BT

Exercise 4.7 | Q 14 | Page 98

If A = [(cos alpha, sin alpha),(-sin alpha, cos alpha)], show that ATA = I, where I is the unit matrix of order 2

Miscellaneous Exercise 4(B) [Pages 99 - 100]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Miscellaneous Exercise 4(B) [Pages 99 - 100]

Miscellaneous Exercise 4(B) | Q I. (1) | Page 99

Select the correct option from the given alternatives:

Given A = [(1, 3),(2, 2)], I = [(1, 0),(0, 1)] if A – λI is a singular matrix then _______

• λ = 0

• λ2 – 3λ – 4 = 0

• λ2 + 3 – 4 = 0

• λ2 – 3λ – 6 = 0

Miscellaneous Exercise 4(B) | Q I. (2) | Page 100

Select the correct option from the given alternatives:

Consider the matrices A = [(4, 6, -1),(3, 0, 2),(1, -2, 5)], B = [(2, 4),(0, 1),(-1, 2)], C = [(3),(1),(2)] out of the given matrix product ________

i) (AB)TC
ii) CTC(AB)
iii) CTAB
iv) ATABBTC

• Exactly one is defined

• Exactly two are defined

• Exactly three are defined

• all four are defined

Miscellaneous Exercise 4(B) | Q I. (3) | Page 100

Select the correct option from the given alternatives:

If A and B are square matrices of equal order, then which one is correct among the following?

• A + B = B + A

• A + B = A – B

• A – B = B – A

• AB = BA

Miscellaneous Exercise 4(B) | Q I. (4) | Page 100

Select the correct option from the given alternatives:

If A = [(1, 2, 2),(2, 1, 2),("a", 2, "b")] is a matrix satisfying the equation AAT = 9I, where I is the identity matrix of order 3, then the ordered pair (a, b) is equal to ________

• (2, –1)

• (–2, 1)

• (2, 1)

• (–2, –1)

Miscellaneous Exercise 4(B) | Q I. (5) | Page 100

Select the correct option from the given alternatives:

If A = [(alpha, 2),(2, alpha)] and |A3| = 125, then α = _______

• ±3

• ±2

• ±5

• 0

Miscellaneous Exercise 4(B) | Q I. (6) | Page 100

Select the correct option from the given alternatives:

If [(5, 7),(x, 1),(2, 6)] - [(1, 2),(-3, 5),(2, y)] = [(4, 5),(4, -4),(0, 4)] then __________

• x = 1, y = –2

• x = –1, y = 2

• x = 1, y = 2

• x = –1, y = –2

Miscellaneous Exercise 4(B) | Q I. (7) | Page 100

Select the correct option from the given alternatives:

If A + B = [(7, 4),(8, 9)] and A − B = [(1, 2),(0, 3)] then the value of A is _______

• [(3, 1),(4, 3)]

• [(4, 3),(4, 6)]

• [(6, 2),(8, 6)]

• [(7, 6),(8, 12)]

Miscellaneous Exercise 4(B) | Q I. (8) | Page 100

Select the correct option from the given alternatives:

If [("x", 3"x" - "y"),("zx" + "z", 3"y" - "w")] = [(3, 2),(4, 7)] then ______

• x = 3, y = 7, z = 1, w = 14

• x = 3, y = −5, z = −1, w = −4

• x = 3, y = 6, z = 2, w = 7

• x = −3, y = –7, z = –1, w = –14

Miscellaneous Exercise 4(B) | Q I. (9) | Page 100

Select the correct option from the given alternatives:

For suitable matrices A, B, the false statement is _____

• (AB)T = ATBT

• (AT)T = A

• (A − B)T = AT − BT

• (A + B)T = AT + BT

Miscellaneous Exercise 4(B) | Q I. (10) | Page 100

Select the correct option from the given alternatives:

If A = [(-2, 1),(0, 3)] and f(x) = 2x2 – 3x, then f(A) = ………

• [(14, 1),(0, -9)]

• [(-14, 1),(0, 9)]

• [(14, -1),(0, 9)]

• [(-14, -1),(0, -9)]

Miscellaneous Exercise 4(B) [Pages 100 - 102]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 4 Determinants and Matrices Miscellaneous Exercise 4(B) [Pages 100 - 102]

Miscellaneous Exercise 4(B) | Q II. (1) (i) | Page 100

If A = diag [2 –3 –5], B = diag [4 –6 –3] and C = diag [–3 4 1] then find B + C – A

Miscellaneous Exercise 4(B) | Q II. (1) (ii) | Page 100

If A = diag [2 –3 –5], B = diag [4 –6 –3] and C = diag [–3 4 1] then find 2A + B – 5C

Miscellaneous Exercise 4(B) | Q II. (2) (i) | Page 101

If f(α) = A = [(cosalpha, -sinalpha, 0),(sinalpha, cosalpha, 0),(0, 0, 1)], Find f(– α)

Miscellaneous Exercise 4(B) | Q II. (2) (ii) | Page 101

If f (α) = A = [(cosalpha, -sinalpha, 0),(sinalpha, cosalpha, 0),(0, 0, 1)], Find f(–α) + f(α)

Miscellaneous Exercise 4(B) | Q II. (3) (i) | Page 101

Find matrices A and B, where 2A – B = [(1, -1),(0, 1)] and A + 3B = [(1, -1),(0, 1)]

Miscellaneous Exercise 4(B) | Q II. (3) (ii) | Page 101

Find matrices A and B, where 3A – B = [(-1, 2, 1),(1, 0, 5)] and A + 5B = [(0, 0, 1),(-1, 0, 0)]

Miscellaneous Exercise 4(B) | Q II. (4) (i) | Page 101

If A = [(2, -3),(3, -2),(-1, 4)], B = [(-3, 4, 1),(2, -1, -3)] Verify (A + BT)T = AT + 2B

Miscellaneous Exercise 4(B) | Q II. (4) (ii) | Page 101

If A = [(2, -3),(3, -2),(-1, 4)], B = [(-3, 4, 1),(2, -1, -3)] Verify (3A - 5BT)T = 3AT – 5B

Miscellaneous Exercise 4(B) | Q II. (5) | Page 101

If A = [(cosalpha, -sinalpha),(sinalpha, cosalpha)] and A + AT = I, where I is unit matrix 2 × 2, then find the value of α

Miscellaneous Exercise 4(B) | Q II. (6) | Page 101

If A = [(1, 2),(3, 2),(-1, 0)] and B = [(1, 3, 2),(4, -1, -3)], show that AB is singular.

Miscellaneous Exercise 4(B) | Q II. (7) | Page 101

If A = [(1, 2, 3),(2, 4, 6),(1, 2, 3)], B = [(1, -1, 1),(-3, 2, -1),(-2, 1, 0)], show that AB and BA are both singular matrices

Miscellaneous Exercise 4(B) | Q II. (8) | Page 101

If A = [(1, -1, 0),(2, 3, 4),(0, 1, 2)], B = [(2, 2, -4),(-4, 2, 4),(2, -1, 5)], show that BA = 6I

Miscellaneous Exercise 4(B) | Q II. (9) | Page 101

If A = [(2, 1),(0, 3)], B = [(1, 2),(3, -2)], verify that |AB| = |A||B|

Miscellaneous Exercise 4(B) | Q II. (10) | Page 101

If = [(cosalpha, sinalpha),(-sinalpha, cosalpha)], show that Aα . Aβ = Aα+β

Miscellaneous Exercise 4(B) | Q II. (11) | Page 101

If A = [(1, omega),(omega^2, 1)], B = [(omega^2, 1),(1, omega)], where ω is a complex cube root of unity, then show that AB + BA + A −2B is a null matrix

Miscellaneous Exercise 4(B) | Q II. (12) | Page 101

If A = [(2, -2, -4),(-1, 3, 4),(1, -2, -3)] show that A2 = A

Miscellaneous Exercise 4(B) | Q II. (13) | Page 101

If A = [(4, -1, -4),(3, 0, -4),(3, -1, -1)], show that A2 = I

Miscellaneous Exercise 4(B) | Q II. (14) | Page 101

If A = [(3, -5),(-4, 2)], show that A2 – 5A – 14I = 0

Miscellaneous Exercise 4(B) | Q II. (15) | Page 101

If A = [(2, -1),(-1, 2)], show that A2 – 4A + 3I = 0

Miscellaneous Exercise 4(B) | Q II. (16) | Page 101

if A = [(-3, 2),(2, -4)], B = [(1, x),(y, 0)], and (A + B)(A – B) = A2 – B2 , find x and y

Miscellaneous Exercise 4(B) | Q II. (17) | Page 102

If A = [(0, 1),(1, 0)] and B = [(0, -1),(1, 0)] show that (A + B)(A – B) ≠ A2 – B

Miscellaneous Exercise 4(B) | Q II. (18) | Page 102

if A = [(2, -1),(3, -2)], find A

Miscellaneous Exercise 4(B) | Q II. (19) (i) | Page 102

Find x, y if, [(0, -1, 4)]{2[(4, 5),(3, 6),(2, -1)] + 3[(4, 3),(1, 4),(0, -1)]} = [(x, y)]

Miscellaneous Exercise 4(B) | Q II. (19) (ii) | Page 102

Find x, y if, {-1 [(1, 2, 1),(2, 0, 3)] + 3[(2, -3, 7),(1, -1, 3)]} [(5),(0),(-1)] = [(x),(y)]

Miscellaneous Exercise 4(B) | Q II. (20) (i) | Page 102

Find x, y, z if {5[(0, 1),(1, 0),(1, 1)] -3[(2, 1),(3, -2),(1, 3)]} [(2),(1)] = [(x - 1),(y + 1),(2z)]

Miscellaneous Exercise 4(B) | Q II. (20) (ii) | Page 102

Find x, y, z if {[(1, 3, 2),(2, 0, 1),(3, 1, 2)] + 2[(3, 0, 2),(1, 4, 5),(2, 1, 0)]} [(1),(2),(3)] = [(x),(y),(z)]

Miscellaneous Exercise 4(B) | Q II. (21) | Page 102

If A = [(2, 1, -3),(0, 2, 6)], B = [(1, 0, -2),(3, -1, 4)], find ABT and ATB

Miscellaneous Exercise 4(B) | Q II. (22) | Page 102

If A = [(2, -4),(3, -2),(0, 1)], B = [(1, -1, 2),(-2, 1, 0)], show that (AB)T = BTAT

Miscellaneous Exercise 4(B) | Q II. (23) | Page 102

If A = [(3, -4),(1, -1)], prove that An = [(1 + 2"n", -4"n"),("n", 1 - 2"n")], for all n ∈ N

Miscellaneous Exercise 4(B) | Q II. (24) | Page 102

If A = [(costheta, sintheta),(-sintheta, costheta)], prove that An = [(cos"n"theta, sin"n"theta),(-sin"n"theta, cos"n"theta)], for all n ∈ N

Miscellaneous Exercise 4(B) | Q II. (25) (i) | Page 102

Two farmers Shantaram and Kantaram cultivate three crops rice,wheat and groundnut. The sale (In Rupees) of these crops by both the farmers for the month of April and may 2008 is given below,

 April sale (In Rs.) Rice Wheat Groundnut Shantaram 15000 13000 12000 Kantaram 18000 15000 8000

 May sale (In Rs.) Rice Wheat Groundnut Shantaram 18000 15000 12000 Kantaram 21000 16500 16000

Find The total sale in rupees for two months of each farmer for each crop

Miscellaneous Exercise 4(B) | Q II. (25) (ii) | Page 102

Two farmers Shantaram and Kantaram cultivate three crops rice,wheat and groundnut. The sale (In Rupees) of these crops by both the farmers for the month of April and may 2008 is given below,

 April sale (In Rs.) Rice Wheat Groundnut Shantaram 15000 13000 12000 Kantaram 18000 15000 8000

 May sale (In Rs.) Rice Wheat Groundnut Shantaram 18000 15000 12000 Kantaram 21000 16500 16000

Find the increase in sales from April to May for every crop of each farmer.

## Chapter 4: Determinants and Matrices

Exercise 4.1Exercise 4.2Exercise 4.3Miscellaneous Exercise 4(A)Exercise 4.4Exercise 4.5Exercise 4.6Exercise 4.7Miscellaneous Exercise 4(B)

## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 4 - Determinants and Matrices

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 4 (Determinants and Matrices) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 4 Determinants and Matrices are Definition and Expansion of Determinants, Minors and Cofactors of Elements of Determinants, Properties of Determinants, Application of Determinants, Cramer’s Rule, Consistency of Three Equations in Two Variables, Area of Triangle and Collinearity of Three Points, Introduction to Matrices, Types of Matrices, Algebra of Matrices, Properties of Matrix Multiplication, Properties of Transpose of a Matrix.

Using Balbharati 11th solutions Determinants and Matrices exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Balbharati Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board 11th prefer Balbharati Textbook Solutions to score more in exam.

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