Chapters
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 Exercise 4.1 [Pages 63 - 64]
Find the value of determinant :
`|(2, -4),(7, -15)|`
Find the value of determinant :
`|(2"i", 3),(4, -"i")|`
Find the value of determinant :
`|(3, -4, 5),(1, 1, -2),(2, 3, 1)|`
Find the value of determinant :
`|("a", "h", "g"),("h", "b", "f"),("g", "f", "c")|`
Find the value of x if
`|(x^2 - x + 1, x + 1),(x + 1, x + 1)|` = 0
Find the value of x if
`|(x, -1, 2),(2x, 1, -3),(3, -4, 5)|` = 29
Find x and y if `|(4"i", "i"^3, 2"i"),(1, 3"i"^2, 4),(5, -3, "i")|` = x + iy where i^{2} = – 1
Find the minor and cofactor of element of the determinant
D = `|(2, -1, 3),(1, 2, -1),(5, 7, 2)|`
Evaluate A = `|(2, -3,5),(6, 0, 4),(1, 5, -7)|` Also find minor and cofactor of elements in the 2^{nd }row of determinant and verify − a_{21}.M_{21} + a_{22}.M_{22} − a_{23}.M_{23} = value of A
where M_{21}, M_{22} , M_{23} are minor of a_{21} , a_{22}, a_{23 } and C_{21}, C_{22}, C_{23} are cofactor of a_{21}, a_{22}, a_{23}
Evaluate A = `|(2, -3,5),(6, 0, 4),(1, 5, -7)|` Also find minor and cofactor of elements in the 2^{nd} row of determinant and verify a_{21} C_{21} + a_{22} C_{22} + a_{23} C_{23} = value of A
where M_{21}, M_{22}, M_{23} are minors of a_{21}, a_{22}, a_{23} and
C_{21}, C_{22}, C_{23} are cofactors of a_{21}, a_{22}, a_{23}.
Find the value of determinant expanding along third column
`|(-1, 1, 2),(-2, 3, -4),(-3, 4, 0)|`
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]
Without expanding evaluate the following determinant:
`|(1, "a", "b" + "c"),(1, "b", "c" + "a"),(1, "c", "a" + "b")|`
Without expanding evaluate the following determinant:
`|(2, 3, 4),(5, 6, 8),(6x, 9x, 12x)|`
Without expanding evaluate the following determinant:
`|(2, 7, 65),(3, 8, 75),(5, 9, 86)|`
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)|`
Using properties of determinant show that
`|("a" + "b", "a", "b"),("a", "a" + "c", "c"),("b", "c", "b" + "c")|` = 4abc
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
Solve the following equation:
`|(x + 2, x + 6, x - 1),(x + 6, x - 1, x + 2),(x - 1, x + 2, x + 6)|` = 0
Solve the following equation:
`|(x -1, x, x - 2),(0, x - 2, x - 3),(0, 0, x - 3)|` = 0
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.
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)|`
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]
Solve the following linear equations by using Cramer’s Rule:
x + y + z = 6, x – y + z = 2, x + 2y – z = 2
Solve the following linear equations by using Cramer’s Rule:
x + y − 2z = –10, 2x + y – 3z = –1, 4x + 6y + z = 2
Solve the following linear equations by using Cramer’s Rule:
x + z = 1, y + z = 1, x + y = 4
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
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.
Examine the consistency of the following equation:
2x − y + 3 = 0, 3x + y − 2 = 0, 11x + 2y − 3 = 0
Examine the consistency of the following equation:
2x + 3y − 4 = 0, x + 2y = 3, 3x + 4y + 5 = 0
Examine the consistency of the following equation:
x + 2y −3 = 0, 7x + 4y − 11 = 0, 2x + 4y − 6 = 0
Find k if the following equations are consistent:
2x + 3y - 2 = 0, 2x + 4y − k = 0, x − 2y + 3k =0
Find k if the following equations are consistent:
kx +3y + 1 = 0, x + 2y + 1 = 0, x + y = 0
Find the area of triangle whose vertices are
A(5, 8), B(5, 0) C(1, 0)
Find the area of triangle whose vertices are
`"P"(3/2, 1), "Q"(4, 2), "R"(4, (-1)/2)`
Find the area of triangle whose vertices are
M(0, 5), N(−2, 3), T(1, −4)
Find the area of quadrilateral whose vertices are
A(−3, 1), B(−2, −2), C(1, 4), D(3, −1)
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"`
Examine the collinearity of the following set of point:
A(3, −1), B(0, −3), C(12, 5)
Examine the collinearity of the following set of point:
P(3, −5), Q(6, 1), R(4, 2)
Examine the collinearity of the following set of point:
`"L"(0, 1/2), "M"(2, -1), "N"(-4, 7/2)`
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]
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 ax^{2} + 2bx + c = 0
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
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
Select the correct option from the given alternatives:
The value of a for which system of equation a^{3}x + (a + 1)^{3} y + (a + 2)^{3}z = 0 ax + (a +1)y + (a + 2)z = 0 and x + y + z = 0 has non zero Soln. is
0
–1
1
2
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)|`
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
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
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
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
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
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]
Answer the following question:
Evaluate `|(2, -5, 7),(5, 2, 1),(9, 0, 2)|`
Answer the following question:
Evaluate `|(1, -3, 12),(0, 2, -4),(9, 7, 2)|`
Answer the following question:
Evaluate determinant along second column
`|(1, -1, 2),(3, 2, -2),(0, 1, -2)|`
Answer the following question:
Evaluate `|(2, 3, 5),(400, 600, 1000),(48, 47, 18)|` by using properties
Answer the following question:
Evaluate `|(101, 102, 103),(106, 107, 108),(1, 2, 3)|` by using properties
Answer the following question:
Find minor and cofactor of elements of the determinant:
`|(-1, 0, 4),(-2, 1, 3),(0, -4, 2)|`
Answer the following question:
Find minor and cofactor of elements of the determinant:
`|(1, -1, 2),(3, 0, -2),(1, 0, 3)|`
Answer the following question:
Find the value of x if
`|(1, 4, 20),(1, -2, -5),(1, 2x, 5x^2)|` = 0
Answer the following question:
Find the value of x if
`|(1, 2x, 4x),(1, 4, 16),(1, 1, 1)| = 0`
Answer the following question:
By using properties of determinant prove that `|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|` = 0
Answer the following question:
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
Answer the following question:
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")|`
Answer the following question:
Without expanding determinant show that
`|(l, "m", "n"),("e", "d", "f"),("u", "v", "w")| = |("n", "f", "w"),(l, "e", "w"),("m", "d", "v")|`
Answer the following question:
Without expanding determinant show that
`|(0, "a", "b"),(-"a", 0, "c"),(-"b", -"c", 0)|` = 0
Answer the following question:
If `|("a", 1, 1),(1, "b", 1),(1, 1, "c")|` = 0 then show that `1/(1 - "a") + 1/(1 - "b") + 1/(1 - "c")` = 1
Answer the following question:
Solve the following linear equations by Cramer’s Rule:
2x − y + z = 1, x + 2y + 3z = 8, 3x + y − 4z =1
Answer the following question:
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`
Answer the following question:
Solve the following linear equations by Cramer’s Rule:
2x+ 3y + 3z = 5 , x − 2y + z = – 4 , 3x – y – 2z = 3
Answer the following question:
Solve the following linear equations by Cramer’s Rule:
x – y + 2z = 7 , 3x + 4y – 5z = 5 , 2x – y + 3z = 12
Answer the following question:
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
Answer the following question:
Find the value of k, if the following equations are consistent:
3x + y − 2 = 0 kx + 2y − 3 = 0 and 2x − y = 3
Answer the following question:
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
Answer the following question:
Find the area of triangle whose vertices are A(−1, 2), B(2, 4), C(0, 0)
Answer the following question:
Find the area of triangle whose vertices are P(3, 6), Q(−1, 3), R(2, −1)
Answer the following question:
Find the area of triangle whose vertices are L(1, 1), M(−2, 2), N(5, 4)
Answer the following question:
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)
Answer the following question:
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)
Answer the following question:
Find the area of quadrilateral whose vertices are A(0, −4), B(4, 0), C(−4, 0), D(0, 4)
Answer the following question:
An amount of â‚¹ 5000 is put into three investments at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is â‚¹ 350. If the combined income from the first two investments is â‚¹ 7000 more than the income from the third. Find the amount of each investment.
Answer the following question:
Show that the lines x − y = 6, 4x − 3y = 20 and 6x + 5y + 8 = 0 are concurrent. Also find the point of concurrence
Answer the following question:
Show that the following points are collinear by determinant:
L(2,5), M(5,7), N(8,9)
Answer the following question:
Show that the following points are collinear by determinant:
P(5,1), Q(1,−1), R(11,4)
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]
Construct a matrix A = = [a_{ij}]_{3×2} whose element a_{ij }is given by
a_{ij} = `(("i" - "j")^2)/(5 - "i")`
Construct a matrix A = = [a_{ij}]_{3×2} whose element a_{ij} is given by
a_{ij} = i – 3j
Construct a matrix A = = [a_{ij}]_{3×2} whose element a_{ij} is given by
a_{ij} = `(("i" + "j")^3)/5`
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)]`
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)]`
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)]`
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]`
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)]`
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)]`
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)]`
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)]`
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)]`
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)]`
Identify the following matrix is singular or non-singular?
`[("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")]`
Identify the following matrix is singular or non-singular?
`[(5, 0, 5),(1, 99, 100),(6, 99, 105)]`
Identify the following matrix is singular or non-singular?
`[(3, 5, 7),(-2, 1, 4),(3, 2, 5)]`
identify the following matrix is singular or non-singular?
`[(7, 5),(-4, 7)]`
Find k if the following matrix is singular:
`[(7, 3),(-2, "k")]`
Find k if the following matrix is singular:
`[(4, 3, 1),(7, "k", 1),(10, 9, 1)]`
Find k if the following matrix is singular:
`[("k" - 1, 2, 3),(3, 1, 2),(1, -2, 4)]`
If A = `[(5, 1, -1),(3, 2, 0)]`, Find (A^{T})^{T}.
If A = `[(7, 3, 1),(-2, -4, 1),(5, 9, 1)]`, Find (A^{T})^{T}.
Find a, b, c if `[(1, 3/5, "a"),("b", -5, -7),(-4, "c", 0)]` is a symmetric matrix.
Find x, y, z If `[(0, -5"i", x),(y, 0, z),(3/2, -sqrt(2), 0)]` is a skew symmetric matrix.
The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:
`[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`
The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:
`[(2, 5, 1),(-5, 4, 6),(-1, -6, 3)]`
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)]`
Construct the matrix A = [a_{ij}]_{3×3} where a_{ij} = i − j. State whether A is symmetric or skew-symmetric.
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]
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
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)
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.
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
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.
Solve the following equations for X and Y, if
3X – Y = `[(1, -1),(-1, 1)] and "X" - 3"Y" = [(0, -1),(0, -1)]`
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)]`
Simplify, `costheta[(costheta, sintheta),(-sintheta, costheta)] + sintheta[(sintheta, -costheta),(costheta, sintheta)]`
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.
Find x and y, if `[(2x + y, -1, 1),(3, 4y, 4)] + [(-1, 6, 4),(3, 0, 3)] = [(3, 5, 5),(6, 18, 7)]`
If = `[(2"a" + "b", 3"a" - "b"),("c" + 2"d", 2"c" - "d")] = [(2, 3),(4, -1)]`, find a, b, c and d.
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.
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
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]
Evaluate: `[(3),(2),(1)][(2,-4,3)]`
Evaluate : `[2 -1 3][(4),(3),(1)]`
If A = `[(1, -3),(4, 2)], "B" = [(4, 1),(3, -2)]` show that AB ≠ BA.
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.
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)]`
Show that AB=BA where,
A = `[(costheta, sintheta),(sintheta, costheta)], "B"[(costheta, -sintheta),(sintheta, costheta)]`
If A = `[(4, 8),(-2, -4)]`, prove that A^{2} = 0
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)]`
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)]`
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)]`
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)]`
If A = `[(1, -2),(5, 6)], "B" = [(3, -1),(3, 7)]`, Find AB-2I, where I is unit matrix of order 2.
If A = `[(4, 3, 2),(-1, 2, 0)], "B" = [(1, 2),(-1, 0),(1, -2)]` show that matrix AB is non singular
If A = `[(1, 2, 0),(5, 4, 2),(0, 7, -3)]`, find the product (A + I)(A − I)
A = `[(alpha, 0),(1, 1)], "B" = [(1, 0),(2, 1)]` find α, if A^{2} = B.
If A = `[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`, Show that A^{2} – 4A is a scalar matrix
If A = `[(1, 0),(-1, 7)]`, find k so that A^{2} – 8A – kI = O, where I is a unit matrix and O is a null matrix of order 2.
If A = `[(8, 4),(10, 5)], "B" = [(5, -4),(10, -8)]` show that (A + B)^{2} = A^{2} + AB + B^{2}
If A = `[(3, 1),(-1, 2)]`, prove that A^{2} – 5A + 7I = 0, where I is unit matrix of order 2
If A = `[(3, 4),(-4, 3)] and "B" = [(2, 1),(-1, 2)]`, show that (A + B)(A – B) = A^{2} – B^{2 }
If A = `[(1, 2),(-1, -2)], "B" = [(2, "a"),(-1, "b")]` and if (A + B)^{2} = A^{2} – B^{2} . find values of a and b
Find matrix X such that AX = B, where A = `[(1, -2),(-2, 1)]` and B = `[(-3),(-1)]`
Find k, if A= `[(3, -2),(4, -2)]` and if A^{2} = kA – 2I
Find x, if `[(1, "x", 1)][(1, 2, 3),(4, 5, 6),(3, 2, 5)] [(1),(-2), (3)]` = 0
Find x and y, if `{4[(2, -1, 3),(1, 0, 2)] -[(3, -3, 4),(2, 1, 1)]} [(2),(-1),(1)] = [(x),(y)]`
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")]`
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, show that `"A"^2=[(cos2alpha, sin2alpha),(-sin2alpha, cos2alpha)]`
If A = `[(1, 2),(3, 5)]` B = `[(0, 4),(2, -1)]`, show that AB ≠ BA, but |AB| = IAl·IBI
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
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]
Find A^{T}, if A = `[(1, 3),(-4, 5)]`
Find A^{T}, if A = `[(2, -6, 1),(-4, 0, 5)]`
If [a_{ij}]_{3x3} where a_{ij} = 2(i – j). Find A and A^{T}. State whether A and A^{T} are symmetric or skew-symmetric matrices?
If A = `[(5, -3),(4, -3),(-2, 1)]`, Prove that (2A)^{T} = 2A^{T}
If A = `[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`, Prove that (3A)^{T} = 3A^{T}
If A = `[(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)]` where i = `sqrt(-1)` Prove that A^{T} = – A
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) = A^{T} + B^{T}
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} = A^{T} – C^{T}
If A = `[(5, 4),(-2, 3)]` and B = `[(-1, 3),(4, -1)]`, then find C^{T} , such that 3A – 2B + C = I, where I is the unit matrix of order 2
If A `[(7, 3, 0),(0, 4, -2)]`, B = `[(0, -2, 3),(2, 1, -4)]` then find A^{T} + 4B^{T}
If A `[(7, 3, 0),(0, 4, -2)]`, B = `[(0, -2, 3),(2, 1, -4)]` then find 5A^{T} – 5B^{T}
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} = A^{T} + 2B^{T} + 3C^{T}
If A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`, prove that (A + B^{T})^{T} = A^{T} + B
Prove that A + A^{T} is a symmetric and A – A^{T} is a skew symmetric matrix, where
A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`
Prove that A + A^{T} is a symmetric and A – A^{T} is a skew symmetric matrix, where
A = `[(5, 2, -4),(3, -7, 2),(4, -5, -3)]`
Express the following matrix as the sum of a symmetric and a skew symmetric matrix
`[(4, -2),(3, -5)]`
Express the following matrix as the sum of a symmetric and a skew symmetric matrix
`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`
If A = `[(2, -1),(3, -2),(4, 1)]` and B = `[(0, 3, -4),(2, -1, 1)]`, verify that (AB)^{T} = B^{T} A^{T}
If A = `[(2, -1),(3, -2),(4, 1)]` and B = `[(0, 3, -4),(2, -1, 1)]`, verify that (BA)^{T} = A^{T} B^{T}
If A = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]`, show that A^{T}A = I, where I is the unit matrix of order 2
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]
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
λ^{2} – 3λ – 4 = 0
λ^{2} + 3 – 4 = 0
λ^{2} – 3λ – 6 = 0
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)^{T}C
ii) C^{T}C(AB)^{T }
iii) C^{T}AB
iv) A^{T}ABB^{T}C
Exactly one is defined
Exactly two are defined
Exactly three are defined
all four are defined
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
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 AA^{T} = 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)
Select the correct option from the given alternatives:
If A = `[(alpha, 2),(2, alpha)]` and |A^{3}| = 125, then α = _______
±3
±2
±5
0
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
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)]`
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
Select the correct option from the given alternatives:
For suitable matrices A, B, the false statement is _____
(AB)^{T} = A^{T}B^{T}
(A^{T})^{T} = A
(A − B)^{T} = A^{T} − B^{T}
(A + B)^{T} = A^{T} + B^{T}
Select the correct option from the given alternatives:
If A = `[(-2, 1),(0, 3)]` and f(x) = 2x^{2} – 3x, then f(A) = ………
`[(14, 1),(0, -9)]`
`[(-14, 1),(0, 9)]`
`[(14, -1),(0, 9)]`
`[(-14, -1),(0, -9)]`
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]
Answer the following question:
If A = diag [2 –3 –5], B = diag [4 –6 –3] and C = diag [–3 4 1] then find B + C – A
Answer the following question:
If A = diag [2 –3 –5], B = diag [4 –6 –3] and C = diag [–3 4 1] then find 2A + B – 5C
Answer the following question:
If f(α) = A = `[(cosalpha, -sinalpha, 0),(sinalpha, cosalpha, 0),(0, 0, 1)]`, Find f(– α)
Answer the following question:
If f (α) = A = `[(cosalpha, -sinalpha, 0),(sinalpha, cosalpha, 0),(0, 0, 1)]`, Find f(–α) + f(α)
Answer the following question:
Find matrices A and B, where 2A – B = `[(1, -1),(0, 1)]` and A + 3B = `[(1, -1),(0, 1)]`
Answer the following question:
Find matrices A and B, where 3A – B = `[(-1, 2, 1),(1, 0, 5)]` and A + 5B = `[(0, 0, 1),(-1, 0, 0)]`
Answer the following question:
If A = `[(2, -3),(3, -2),(-1, 4)]`, B = `[(-3, 4, 1),(2, -1, -3)]` Verify (A + B^{T})^{T} = A^{T} + 2B
Answer the following question:
If A = `[(2, -3),(3, -2),(-1, 4)]`, B = `[(-3, 4, 1),(2, -1, -3)]` Verify (3A - 5B^{T})^{T} = 3A^{T} – 5B
Answer the following question:
If A = `[(cosalpha, -sinalpha),(sinalpha, cosalpha)]` and A + A^{T} = I, where I is unit matrix 2 × 2, then find the value of α
Answer the following question:
If A = `[(1, 2),(3, 2),(-1, 0)]` and B = `[(1, 3, 2),(4, -1, -3)]`, show that AB is singular.
Answer the following question:
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
Answer the following question:
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
Answer the following question:
If A = `[(2, 1),(0, 3)]`, B = `[(1, 2),(3, -2)]`, verify that |AB| = |A||B|
Answer the following question:
If = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, show that A_{α} . A_{β} = A_{α+β}
Answer the following question:
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
Answer the following question:
If A = `[(2, -2, -4),(-1, 3, 4),(1, -2, -3)]` show that A^{2} = A
Answer the following question:
If A = `[(4, -1, -4),(3, 0, -4),(3, -1, -1)]`, show that A^{2} = I
Answer the following question:
If A = `[(3, -5),(-4, 2)]`, show that A^{2} – 5A – 14I = 0
Answer the following question:
If A = `[(2, -1),(-1, 2)]`, show that A^{2} – 4A + 3I = 0
Answer the following question:
if A = `[(-3, 2),(2, -4)]`, B = `[(1, x),(y, 0)]`, and (A + B)(A – B) = A^{2} – B^{2} , find x and y
Answer the following question:
If A = `[(0, 1),(1, 0)]` and B = `[(0, -1),(1, 0)]` show that (A + B)(A – B) ≠ A^{2} – B^{2 }
Answer the following question:
if A = `[(2, -1),(3, -2)]`, find A^{3 }
Answer the following question:
Find x, y if, `[(0, -1, 4)]{2[(4, 5),(3, 6),(2, -1)] + 3[(4, 3),(1, 4),(0, -1)]} = [(x, y)]`
Answer the following question:
Find x, y if, `{-1 [(1, 2, 1),(2, 0, 3)] + 3[(2, -3, 7),(1, -1, 3)]} [(5),(0),(-1)] = [(x),(y)]`
Answer the following question:
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)]`
Answer the following question:
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)]`
Answer the following question:
If A = `[(2, 1, -3),(0, 2, 6)]`, B = `[(1, 0, -2),(3, -1, 4)]`, find AB^{T} and A^{T}B
Answer the following question:
If A = `[(2, -4),(3, -2),(0, 1)]`, B = `[(1, -1, 2),(-2, 1, 0)]`, show that (AB)^{T} = B^{T}A^{T}
Answer the following question:
If A = `[(3, -4),(1, -1)]`, prove that A^{n} = `[(1 + 2"n", -4"n"),("n", 1 - 2"n")]`, for all n ∈ N
Answer the following question:
If A = `[(costheta, sintheta),(-sintheta, costheta)]`, prove that A^{n} = `[(cos"n"theta, sin"n"theta),(-sin"n"theta, cos"n"theta)]`, for all n ∈ N
Answer the following question:
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
Answer the following question:
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
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, Cramerâ€™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.
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