# Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 2 - Matrices [Latest edition]

## Chapter 2: Matrices

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Exercise 2.5Exercise 2.6Miscellaneous Exercise 2
Exercise 2.1 [Pages 39 - 40]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 2 MatricesExercise 2.1 [Pages 39 - 40]

Exercise 2.1 | Q 1.1 | Page 39

Construct  a matrix A = [a_("ij")]_(3 xx 2) whose element aij is given by

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

Exercise 2.1 | Q 1.2 | Page 39

Construct a matrix A = [a_("ij")]_(3 xx 2) whose element aij is given by

aij = i – 3j

Exercise 2.1 | Q 1.3 | Page 39

Construct a matrix A = [a_("ij")]_(3 xx 2) whose element aij is given by

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

Exercise 2.1 | Q 2.1 | Page 39

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

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

Exercise 2.1 | Q 2.2 | Page 39

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

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

Exercise 2.1 | Q 2.3 | Page 39

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

[9   sqrt(2)   -3]

Exercise 2.1 | Q 2.4 | Page 39

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

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

Exercise 2.1 | Q 2.5 | Page 39

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

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

Exercise 2.1 | Q 2.6 | Page 39

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

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

Exercise 2.1 | Q 2.7 | Page 39

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

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

Exercise 2.1 | Q 3.1 | Page 39

Which of the following matrices are singular or non singular?

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

Exercise 2.1 | Q 3.2 | Page 39

Which of the following matrices are singular or non singular?

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

Exercise 2.1 | Q 3.3 | Page 40

Which of the following matrices are singular or non singular?

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

Exercise 2.1 | Q 3.4 | Page 40

Which of the following matrices are singular or non singular?

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

Exercise 2.1 | Q 4.1 | Page 40

Find K if the following matrices are singular.

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

Exercise 2.1 | Q 4.2 | Page 40

Find K if the following matrices are singular.

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

Exercise 2.1 | Q 4.3 | Page 40

Find K if the following matrices are singular.

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

Exercise 2.2 [Pages 46 - 47]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 2 MatricesExercise 2.2 [Pages 46 - 47]

Exercise 2.2 | Q 1.1 | Page 46

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 2.2 | Q 1.2 | Page 46

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 2.2 | Q 2 | Page 46

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 2.2 | Q 3 | Page 46

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 2.2 | Q 4 | Page 46

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 2.2 | Q 5 | Page 46

If A = [(5, 1, -4),(3, 2, 0)], find (AT)T.

Exercise 2.2 | Q 6 | Page 46

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

Exercise 2.2 | Q 7 | Page 47

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

Exercise 2.2 | Q 8 | Page 47

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

Exercise 2.2 | Q 9.1 | Page 47

For each of the following matrices, find its transpose and state whether it is symmetric, skew- symmetric or neither.

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

Exercise 2.2 | Q 9.2 | Page 47

For each of the following matrices, find its transpose and state whether it is symmetric, skew- symmetric or neither.

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

Exercise 2.2 | Q 9.3 | Page 47

For each of the following matrices, find its transpose and 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 2.2 | Q 10 | Page 47

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

Exercise 2.2 | Q 11 | Page 47

Solve the following equations for X and Y, if 3X − Y = [(1, -1),(-1, 1)]  and X – 3Y = [(0, -1),(0, -1)].

Exercise 2.2 | Q 12 | Page 47

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

Exercise 2.2 | Q 13 | Page 47

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 2.2 | Q 14 | Page 47

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

Exercise 2.2 | Q 15.1 | Page 47

There are two book shops own 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)][("Suresh"), ("Ganesh")]

August Sales (in Rupees :

B = [(6650, 7055, 8905),(7000, 7500, 10200)][("Suresh"), ("Ganesh")]

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

Exercise 2.2 | Q 15.2 | Page 47

There are two book shops own 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)][("Suresh"), ("Ganesh")]
August Sales (in Rupees :
B = [(6650, 7055, 8905),(7000, 7500, 10200)][("Suresh"), ("Ganesh")]
If both book shops get 10% profit in the month of August 2017, find the profit for each book seller in each subject in that month.

Exercise 2.3 [Pages 55 - 56]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 2 MatricesExercise 2.3 [Pages 55 - 56]

Exercise 2.3 | Q 1.1 | Page 55

Evaluate : [(3),(2),(1)][2  -4   3]

Exercise 2.3 | Q 1.2 | Page 55

Evaluate : [2 - 1   3][(4),(3),(1)]

Exercise 2.3 | Q 2 | Page 55

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 2.3 | Q 3 | Page 55

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 2.3 | Q 4 | Page 55

Verify A(BC) = (AB)C, if A = [(1, 0, 1),(2, 3, 0),(0, 4, 5)], "B" = [(2, -2),(-1, 1),(0, 3)]

Exercise 2.3 | Q 5 | Page 55

Verify that A(B + C) = AB + AC, if A = [(4, -2),(2, 3)], "B" = [(-1, 1),(3, -2)] " and C" = [(4 ,1),(2, -1)].

Exercise 2.3 | Q 6 | Page 56

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

Exercise 2.3 | Q 7 | Page 56

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

Exercise 2.3 | Q 8 | Page 56

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

Exercise 2.3 | Q 9 | Page 56

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

Exercise 2.3 | Q 10 | Page 56

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

Exercise 2.3 | Q 11 | Page 56

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

Exercise 2.3 | Q 12 | Page 56

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

Exercise 2.3 | Q 13 | Page 56

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

Exercise 2.3 | Q 14 | Page 56

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

Exercise 2.3 | Q 15 | Page 56

Jay and Ram are two friends. Jay wants to buy 4 pens and 8 notebooks, Ram wants to buy 5 pens and 12 notebooks. The price of one pen and one notebook was ₹ 6 and ₹ 10 respectively. Using matrix multiplication, find the amount each one of them requires for buying the pens and notebooks.

Exercise 2.4 [Pages 59 - 60]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 2 MatricesExercise 2.4 [Pages 59 - 60]

Exercise 2.4 | Q 1.1 | Page 59

Find AT,  if A = [(1, 3),(-4, 5)]

Exercise 2.4 | Q 1.2 | Page 59

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

Exercise 2.4 | Q 2 | Page 59

If [aij]3×3, where aij = 2(i – j), find A and AT. State whether A and AT both are symmetric or skew-symmetric matrices?

Exercise 2.4 | Q 3 | Page 59

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

Exercise 2.4 | Q 4 | Page 59

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

Exercise 2.4 | Q 5.1 | Page 59

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

Exercise 2.4 | Q 5.2 | Page 59

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

Exercise 2.4 | Q 6 | Page 59

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

Exercise 2.4 | Q 7.1 | Page 59

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

Exercise 2.4 | Q 7.2 | Page 59

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

Exercise 2.4 | Q 8 | Page 59

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 + 3C)T = AT + 2BT + CT.

Exercise 2.4 | Q 9 | Page 59

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

Exercise 2.4 | Q 10.1 | Page 59

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 2.4 | Q 10.2 | Page 59

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 2.4 | Q 11.1 | Page 59

Express each of the following matrix as the sum of a symmetric and a skew symmetric matrix [(4, -2),(3, -5)].

Exercise 2.4 | Q 11.2 | Page 59

Express each of the following matrix as the sum of a symmetric and a skew symmetric matrix [(3, 3, -1),(-2, -2, 1),(-4, -5, 2)].

Exercise 2.4 | Q 12.1 | Page 60

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

Exercise 2.4 | Q 12.2 | Page 60

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

Exercise 2.5 [Pages 71 - 72]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 2 MatricesExercise 2.5 [Pages 71 - 72]

Exercise 2.5 | Q 1.1 | Page 71

Apply the given elementary transformation on each of the following matrices [(3, -4),(2, 2)], R1 ↔ R2.

Exercise 2.5 | Q 1.2 | Page 71

Apply the given elementary transformation on each of the following matrices [(2, 4),(1, -5)], C1 ↔ C2.

Exercise 2.5 | Q 1.3 | Page 71

Apply the given elementary transformation on each of the following matrices [(3, 1, -1),(1, 3, 1),(-1, 1, 3)], 3R2 and C2 ↔ C2 – 4C1.

Exercise 2.5 | Q 2 | Page 71

Transform [(1, -1, 2),(2, 1, 3),(3, 2, 4)] into an upper traingular matrix by suitable row transformations.

Exercise 2.5 | Q 3.1 | Page 72

Find the cofactor matrix, of the following matrices : [(1, 2),(5, -8)]

Exercise 2.5 | Q 3.2 | Page 72

Find the cofactor matrix, of the following matrices: [(5, 8, 7),(-1, -2, 1),(-2, 1, 1)]

Exercise 2.5 | Q 4.1 | Page 72

Find the adjoint of the following matrices : [(2, -3),(3, 5)]

Exercise 2.5 | Q 4.2 | Page 72

Fid the adjoint of the following matrices : [(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]

Exercise 2.5 | Q 5.1 | Page 72

Find the inverse of the following matrices by the adjoint method [(3, -1),(2, -1)].

Exercise 2.5 | Q 5.2 | Page 72

Find the inverse of the following matrices by the adjoint method [(2, -2),(4, 5)].

Exercise 2.5 | Q 5.3 | Page 72

Find the inverse of the following matrices by the adjoint method [(1, 2, 3),(0, 2, 4),(0, 0, 5)].

Exercise 2.5 | Q 6.1 | Page 72

Find the inverse of the following matrices by transformation method: [(1, 2),(2, -1)]

Exercise 2.5 | Q 6.2 | Page 72

Find the inverse of the following matrices by transformation method: [(2, 0, -1),(5, 1, 0),(0, 1, 3)]

Exercise 2.5 | Q 7 | Page 72

Find the inverse of  A = [(1, 0, 1),(0, 2, 3),(1, 2, 1)] by elementary column transformations.

Exercise 2.5 | Q 8 | Page 72

Find the inverse [(1, 2, 3 ),(1, 1, 5),(2, 4, 7)] of  the elementary row tranformation.

Exercise 2.5 | Q 9 | Page 72

If A = [(1, 0, 1),(0, 2, 3),(1, 2, 1)] "and B" = [(1, 2, 3),(1, 1, 5),(2, 4, 7)], then find a matrix X such that XA = B.

Exercise 2.5 | Q 10 | Page 72

Find matrix X, if AX = B, where A = [(1, 2, 3),(-1, 1, 2),(1, 2, 4)] "and B" = [(1),(2),(3)].

Exercise 2.6 [Pages 79 - 80]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 2 MatricesExercise 2.6 [Pages 79 - 80]

Exercise 2.6 | Q 1.1 | Page 79

Solve the following equations by method of inversion.
x + 2y = 2, 2x + 3y = 3

Exercise 2.6 | Q 1.2 | Page 79

Solve the following equations by method of inversion.
2x + y = 5, 3x + 5y = – 3

Exercise 2.6 | Q 1.3 | Page 79

Solve the following equations by method of inversion.
2x – y + z = 1, x + 2y + 3z = 8 and 3x + y – 4z = 1

Exercise 2.6 | Q 1.4 | Page 79

Solve the following equations by method of inversion.
x + y + z = 1, x – y + z = 2 and x + y – z = 3

Exercise 2.6 | Q 2.1 | Page 80

Express the following equations in matrix form and solve them by method of reduction.
x + 3y  = 2, 3x + 5y = 4

Exercise 2.6 | Q 2.2 | Page 80

Express the following equations in matrix form and solve them by method of reduction.
3x – y = 1, 4x + y = 6

Exercise 2.6 | Q 2.3 | Page 80

Express the following equations in matrix form and solve them by method of reduction.
x + 2y + z = 8, 2x + 3y – z = 11 and 3x – y – 2z = 5

Exercise 2.6 | Q 2.4 | Page 80

Express the following equations in matrix form and solve them by method of reduction.
x + y + z = 1, 2x + 3y + 2z = 2 and x + y + 2z = 4

Exercise 2.6 | Q 3 | Page 80

The total cost of 3 T.V. and 2 V.C.R. is ₹ 35,000. The shopkeeper wants profit of ₹1000 per television and ₹ 500 per V.C.R. He can sell 2 T.V. and 1 V.C.R. and get the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. and a V.C.R.

Exercise 2.6 | Q 4 | Page 80

The sum of the cost of one Economic book, one Co-operation book and one account book is ₹ 420. The total cost of an Economic book, 2 Co-operation books and an Account book is ₹ 480. Also the total cost of an Economic book, 3 Co-operation books and 2 Account books is ₹ 600. Find the cost of each book.

Miscellaneous Exercise 2 [Pages 81 - 86]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 2 MatricesMiscellaneous Exercise 2 [Pages 81 - 86]

Miscellaneous Exercise 2 | Q 1.01 | Page 81

Choose the correct alternative.

If AX = B, where A = [(-1, 2),(2, -1)], "B" = [(1),(1)], then X = _______

• [(3/5),(3/7)]

• [(7/3),(5/3)]

• [(1),(1)]

• [(1),(2)]

Miscellaneous Exercise 2 | Q 1.02 | Page 81

Choose the correct alternative.

The matrix [(8, 0, 0),(0, 8, 0),(0, 0, 8)] is _______

• identity matrix

• scalar matrix

• null matrix

• diagonal matrix

Miscellaneous Exercise 2 | Q 1.03 | Page 81

Choose the correct alternative.

The matrix [(0, 0, 0),(0, 0, 0)] is _______

• identity matrix

• diagonal matrix

• scalar matrix

• null matrix

Miscellaneous Exercise 2 | Q 1.04 | Page 81

Choose the correct alternative.

If A = [("a", 0, 0),(0, "a", 0),(0, 0,"a")], then |adj.A| = _______

• a12

• a9

• a6

• a–3

Miscellaneous Exercise 2 | Q 1.05 | Page 81

Choose the correct alternative.

Adjoint of [(2, -3),(4, -6)] is _______

• [(-6, 3),(-4, 2)]

• [(6, 3),(-4, 2)]

• [(-6, -3),(4, 2)]

• [(-6, 3),(4, -2)]

Miscellaneous Exercise 2 | Q 1.06 | Page 82

Choose the correct alternative.

If A = diag [d1, d2, d3,...,dn], where di ≠ 0, for i = 1, 2, 3,...,n, then A–1 = _______

• "diag".[1/"d"_1, 1/"d"_2, 1/"d"_3,......,1/"d"_"n"],

• D

• I

• O

Miscellaneous Exercise 2 | Q 1.07 | Page 82

Choose the correct alternative.

If A2 + mA + nI = O and n ≠ 0, |A| ≠ 0, then A–1 = _______

• (-1)/"m"("A" + "nI")

• (-1)/"n"("A" + "mI")

• (-1)/"n"("I" + "mA")

• (A + mnI)

Miscellaneous Exercise 2 | Q 1.08 | Page 82

Choose the correct alternative.

If a 3 x 3 matrix B has it inverse equal to B, thenB2 = _______

• [(0, 1, 1),(0, 1, 0),(1, 0, 1)]

• [(1, 1, 1),(1, 1, 1),(1, 0, 1)]

• [(1, 0, 1),(0, 1, 0),(0, 0, 0)]

• [(1, 0, 0),(0, 1, 0),(0, 0, 1)]

Miscellaneous Exercise 2 | Q 1.09 | Page 82

Choose the correct alternative.

If A = |(alpha, 4),(4, alpha)| and |A3| = 729, then α =

• ±3

• ±4

• ±5

• ±6

Miscellaneous Exercise 2 | Q 1.1 | Page 82

Choose the correct alternative.

If A and B are square matrices of order n × n such that A2 – B2 = (A – B)(A + B), then which of the following will be always true?

• AB = BA

• either of A or B is a zero matrix

• either of A and B is an identity matrix

• A = B

Miscellaneous Exercise 2 | Q 1.11 | Page 82

Choose the correct alternative.

If A = [(2, 5),(1, 3)], then A–1 = _______

• [(3, -5),(1, 2)]

• [(3, -5),(-1, 2)]

• [(3, 5),(-1, 2)]

• [(3, -5),(1, -2)]

Miscellaneous Exercise 2 | Q 1.12 | Page 82

Choose the correct alternative.

If A is a 2 x 2 matrix such that A(adj. A) = [(5, 0),(0, 5)], then |A| = _______

• 0

• 5

• 10

• 25

Miscellaneous Exercise 2 | Q 1.13 | Page 82

If A is a no singular matrix, then det (A–1) = _______

• 1

• 0

• det(A)

• 1/("det"("A")

Miscellaneous Exercise 2 | Q 1.14 | Page 82

If A = [(1, 2),(-3, -1)], "B" = [(-1, 0),(1, 5)], then AB =

• [(1, -10),(1, 20)]

• [(1, 10),(-1, 20)]

• [(1, 10),(2,- 5)]

• [(1, 10),(-1, -20)]

Miscellaneous Exercise 2 | Q 1.15 | Page 82

Choose the correct alternative.

If x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6, then (y, z) = _______

• (–1, 0)

• (1, 0)

• (1, –1)

• (–1, 1)

Miscellaneous Exercise 2 | Q 2.01 | Page 83

Fill in the blank:

A = [(3),(1)] is ........................ matrix.

Miscellaneous Exercise 2 | Q 2.02 | Page 83

Fill in the blank :

Order of matrix [(2, 1, 1),(5, 1, 8)] is _______

Miscellaneous Exercise 2 | Q 2.03 | Page 83

Fill in the blank :

If A = [(4, x),(6, 3)] is a singular matrix, then x is _______

Miscellaneous Exercise 2 | Q 2.04 | Page 83

Fill in the blank :

Matrix B = [(0, 3, 1),(-3, 0, -4),("p", 4, 0)] is skew symmetric, then the value of p is _______

Miscellaneous Exercise 2 | Q 2.05 | Page 83

Fill in the blank :

If A = [aij]2x3 and B = [bij]mx1 and AB is defined, then m = _______

Miscellaneous Exercise 2 | Q 2.06 | Page 83

Fill in the blank :

If A = [(3, -5),(2, 5)], then co-factor of a12 is _______

Miscellaneous Exercise 2 | Q 2.07 | Page 83

Fill in the blank :

If A = [aij]mxm is a non-singular matrix, then A–1 = (1)/(......) adj(A).

Miscellaneous Exercise 2 | Q 2.08 | Page 83

Fill in the blank :

(AT)T = _______

Miscellaneous Exercise 2 | Q 2.09 | Page 83

Fill in the blank :

If A = [(2, 1),(1, 1)] "and" "A"^-1 = [(1, 1),(x, 2)], then x = _______

Miscellaneous Exercise 2 | Q 2.1 | Page 83

Fill in the blank :

If a1x + b1y = c1 and a2x + b2y = c2, then matrix form is [(......, ......),(......, ......)] = [(x),(y)] = [(......),(......)]

Miscellaneous Exercise 2 | Q 3.01 | Page 83

State whether the following is True or False :

Single element matrix is row as well as column matrix.

• True

• False

Miscellaneous Exercise 2 | Q 3.02 | Page 83

State whether the following is True or False :

Every scalar matrix is unit matrix.

• True

• False

Miscellaneous Exercise 2 | Q 3.03 | Page 83

State whether the following is True or False :

A = [(4, 5),(6, 1)] is no singular matrix.

• True

• False

Miscellaneous Exercise 2 | Q 3.04 | Page 83

State whether the following is True or False :

If A is symmetric, then A = –AT.

• True

• False

Miscellaneous Exercise 2 | Q 3.05 | Page 83

State whether the following is True or False :

If AB and BA both exist, then AB = BA.

• True

• False

Miscellaneous Exercise 2 | Q 3.06 | Page 83

State whether the following is True or False :

If A and B are square matrices of same order, then (A + B)2 = A2 + 2AB + B2.

• True

• False

Miscellaneous Exercise 2 | Q 3.07 | Page 83

State whether the following is True or False :

If A and B are conformable for the product AB, then (AB)T = ATBT.

• True

• False

Miscellaneous Exercise 2 | Q 3.08 | Page 83

State whether the following is True or False :

Singleton matrix is only row matrix.

• True

• False

Miscellaneous Exercise 2 | Q 3.09 | Page 83

State whether the following is True or False :

A = [(2, 1),(10, 5)] is invertible matrix.

• True

• False

Miscellaneous Exercise 2 | Q 3.1 | Page 83

State whether the following is True or False :

A(adj. A) = |A| I, where I is the unit matrix.

• True

• False

Miscellaneous Exercise 2 | Q 4.01 | Page 84

Solve the following :

Find k, if [(7, 3),(5, "k")] is a singular matrix.

Miscellaneous Exercise 2 | Q 4.02 | Page 84

Solve the following :

Find x, y, z if [(2, x, 5),(3, 1, z),(y, 5, 8)] is a symmetric matrix.

Miscellaneous Exercise 2 | Q 4.03 | Page 84

Solve the following :

If A = [(1, 5),(7, 8),(9, 5)], "B" = [(2, 4),(1, 5),(-8, 6)] "C" = [(-2, 3),(1, -5),(7, 8)] then show that (A + B) + C = A + (B + C).

Miscellaneous Exercise 2 | Q 4.04 | Page 84

Solve the following :

If A = [(2, 5),(3, 7)], "B" = 4[(1, 7),(-3, 0)], find matrix A – 4B + 7I, where I is the unit matrix of order 2.

Miscellaneous Exercise 2 | Q 4.05 | Page 84

Solve the following :

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

Miscellaneous Exercise 2 | Q 4.05 | Page 84

Solve the following :

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

Miscellaneous Exercise 2 | Q 4.06 | Page 84

Solve the following :

If A = [(1, 2, 3),(2, 4, 6),(1, 2, 3)],"B" = [(1, -1, 1),(-3, 2, -1),(-2, 1, 0)], then show that AB and BA are bothh singular martices.

Miscellaneous Exercise 2 | Q 4.07 | Page 84

Solve the following :

If A = [(3, 1),(1, 5)], "B" = [(1, 2),(5, -2)], verify |AB| = |A| |B|.

Miscellaneous Exercise 2 | Q 4.08 | Page 84

Solve the following :

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

Miscellaneous Exercise 2 | Q 4.09 | Page 84

Solve the following :

If A = [(-3, 2),(2, 4)], "B" = [(1, "a"), ("b", 0)] and (A + B) (A – B) = A2 – B2, find a and b.

Miscellaneous Exercise 2 | Q 4.1 | Page 84

Solve the following :

if A = [(1, 2),(-1, 3)], then find A3.

Miscellaneous Exercise 2 | Q 4.11 | Page 84

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

Miscellaneous Exercise 2 | Q 4.12 | Page 84

Solve the following :

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

Miscellaneous Exercise 2 | Q 4.13 | Page 85

Solve the following :

If A = [(1, 0, 0),(2, 1, 0),(3, 3, 1)], the reduce it to unit matrix by using row transformations.

Miscellaneous Exercise 2 | Q 4.14 | Page 85

Solve the following :

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 2016 is given below,

 April 2016 (in ₹.) Rice Wheat Groundnut Shantaram 15000 13000 12000 Kantaram 18000 15000 8000
 May 2016 (in ₹.) 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 2 | Q 4.14 | Page 85

Solve the following :

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 2016 is given below,

 April 2016 (in ₹.) Rice Wheat Groundnut Shantaram 15000 13000 12000 Kantaram 18000 15000 8000
 May 2016 (in ₹.) Rice Wheat Groundnut Shantaram 18000 15000 12000 Kantaram 21000 16500 16000

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

Miscellaneous Exercise 2 | Q 4.15 | Page 85

Check whether the following matrices are invertible or not:

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

Miscellaneous Exercise 2 | Q 4.15 | Page 85

Check whether the following matrices are invertible or not:

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

Miscellaneous Exercise 2 | Q 4.15 | Page 85

Check whether the following matrices are invertible or not:

[(3, 4, 5),(1, 1, 0),(1, 4, 5)]

Miscellaneous Exercise 2 | Q 4.15 | Page 85

Check whether the following matrices are invertible or not:

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

Miscellaneous Exercise 2 | Q 4.16 | Page 85

Find inverse of the following matrices (if they exist) by elementary transformations :

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

Miscellaneous Exercise 2 | Q 4.16 | Page 85

Find inverse of the following matrices (if they exist) by elementary transformations :

[(2, 1),(7, 4)]

Miscellaneous Exercise 2 | Q 4.16 | Page 85

Find inverse of the following matrices (if they exist) by elementary transformations :

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

Miscellaneous Exercise 2 | Q 4.16 | Page 85

Find inverse of the following matrices (if they exist) by elementary transformations :

[(2, 0, -1),(5, 1, 0),(0, 1, 3)]

Miscellaneous Exercise 2 | Q 4.17 | Page 85

Find the inverse of [(3, 1, 5),(2, 7, 8),(1, 2, 5)] by adjoint method.

Miscellaneous Exercise 2 | Q 4.18 | Page 85

Solve the following equations by method of inversion :

4x – 3y – 2 = 0, 3x – 4 + 6 = 0

Miscellaneous Exercise 2 | Q 4.18 | Page 85

Solve the following equations by method of inversion : x + y – z = 2, x – 2y + z = 3 and 2x – y – 3z = – 1

Miscellaneous Exercise 2 | Q 4.18 | Page 85

Solve the following equations by method of inversion : x – y + z = 4, 2x + y –z = 0 , x + y + z = 2

Miscellaneous Exercise 2 | Q 4.19 | Page 85

Solve the following equations by method of reduction : 2x + y = 5, 3x + 5y = – 3

Miscellaneous Exercise 2 | Q 4.19 | Page 85

Solve the following equations by method of reduction :

x + 2y – z = 3 , 3x – y + 2z = 1 and 2x – 3y + 3z = 2

Miscellaneous Exercise 2 | Q 4.19 | Page 85

Solve the following equations by method of reduction :

x – 3y + z = 2 , 3x + y + z = 1 and 5x + y + 3z = 3

Miscellaneous Exercise 2 | Q 4.2 | Page 86

The sum of three numbers is 6. If we multiply third number by 3 and add it to the second number we get 11. By adding the first and third number we get a number which is double the second number. Use this information and find a system of linear equations. Find the three numbers using matrices.

## Chapter 2: Matrices

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Exercise 2.5Exercise 2.6Miscellaneous Exercise 2

## Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 2 - Matrices

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