# NCERT solutions for Mathematics Class 12 chapter 3 - Matrices [Latest edition]

## Solutions for Chapter 3: Matrices

Below listed, you can find solutions for Chapter 3 of CBSE, Karnataka Board PUC NCERT for Mathematics Class 12.

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5
Exercise 3.1 [Pages 64 - 65]

### NCERT solutions for Mathematics Class 12 Chapter 3 Matrices Exercise 3.1 [Pages 64 - 65]

Exercise 3.1 | Q 1.1 | Page 64

In the matrix A = [(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)]

The order of the matrix

Exercise 3.1 | Q 1.2 | Page 80

Let A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]

Find A - B

Exercise 3.1 | Q 1.2 | Page 64

In the matrix A = [(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)]

The number of elements,

Exercise 3.1 | Q 1.3 | Page 64

In the matrix A = [(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)] Write the elements a13a21a33a24a23

Exercise 3.1 | Q 2 | Page 64

If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?

Exercise 3.1 | Q 3 | Page 64

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Exercise 3.1 | Q 4.1 | Page 64

Construct a 2 × 2 matrix, A = [a_(ij)], whose elements are given by:

a_(ij) = (i+j)^2/2

Exercise 3.1 | Q 4.2 | Page 64

Construct a 2 × 2 matrix, A= [a_(ij)], whose elements are given by a_(ij) = i/j

Exercise 3.1 | Q 4.3 | Page 64

Construct a 2 × 2 matrix, A = [a_(ij)]  whose elements are given by:

a_(ij) = (1 + 2j)^2/2

Exercise 3.1 | Q 5.1 | Page 64

Construct a 3 × 4 matrix, whose elements are given by a_(ij) = 1/2 |-3i + j|

Exercise 3.1 | Q 5.2 | Page 64

Construct a 3 × 4 matrix, whose elements are given by a_(ij) = 2i - j

Exercise 3.1 | Q 6.1 | Page 64

Find the value of xy, and z from the following equation:

[(4,3),(x,5)] = [(y,z),(1,5)]

Exercise 3.1 | Q 6.2 | Page 64

Find the value of xy, and z from the following equation:

[(x+y, 2),(5+z, xy)] = [(6,2), (5,8)]

Exercise 3.1 | Q 6.3 | Page 64

Find the value of xy, and z from the following equation:

[(x+y+z), (x+z), (y+z)] = [(9),(5),(7)]

Exercise 3.1 | Q 7 | Page 64

Find the value of abc, and d from the equation:

[(a-b, 2a+c),(2a-b, 3x+d)] = [(-1,5),(0,13)]

Exercise 3.1 | Q 8 | Page 65

A = [a_(ij)]_(mxxn) is a square matrix, if

(A) m < n

(B) m > n

(C) m = n

(D) None of these

Exercise 3.1 | Q 9 | Page 65

Which of the given values of x and y make the following pair of matrices equal

[(3x+7, 5),(y+1, 2-3x)] = [(0,y-2),(8,4)]

(A) x= (-1)/3, y = 7

(B) Not possible to find

(C) y = 7, x = (-2)/3

(D) x = (-1)/3, y = (-2)/3

Exercise 3.1 | Q 10 | Page 65

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

(A) 27

(B) 18

(C) 81

(D) 512

Exercise 3.2 [Pages 80 - 83]

### NCERT solutions for Mathematics Class 12 Chapter 3 Matrices Exercise 3.2 [Pages 80 - 83]

Exercise 3.2 | Q 1.1 | Page 80

Let A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]

Find  A + B

Exercise 3.2 | Q 1.3 | Page 80

Let A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]

Find  3A -C

Exercise 3.2 | Q 1.4 | Page 80

Let A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]   Find AB

Exercise 3.2 | Q 1.5 | Page 80

Let A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]

Find BA

Exercise 3.2 | Q 2.1 | Page 80

Compute the following: [(a,b),(-b, a)] + [(a,b),(b,a)]

Exercise 3.2 | Q 2.2 | Page 80

Compute the following:

[(a^2+b^2, b^2+c^2),(a^2+c^2, a^2+b^2)] + [(2ab , 2bc),(-2ac, -2ab)]

Exercise 3.2 | Q 2.3 | Page 80

Compute the following:

[(-1,4, -6),(8,5,16),(2,8,5)] + [(12,7,6),(8,0,5),(3,2,4)]

Exercise 3.2 | Q 2.5 | Page 80

Compute the following:

[(cos^2x, sin^2 x),(sin^2 x ,cos^2 x)]+[(sin^2 x, cos^2 x), (cos^2 x, sin^2 x)]

Exercise 3.2 | Q 3.1 | Page 80

Compute the indicated products

[(a,b),(-b,a)][(a,-b),(b,a)]

Exercise 3.2 | Q 3.2 | Page 80

Compute the indicated products

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

Exercise 3.2 | Q 3.3 | Page 80

Compute the indicated products

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

Exercise 3.2 | Q 3.4 | Page 80

Compute the indicated products

[(2,3,4),(3,4,5),(4,5,6)][(1,-3,5),(0,2,4), (3,0,5)]

Exercise 3.2 | Q 3.5 | Page 80

Compute the indicated products

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

Exercise 3.2 | Q 3.6 | Page 80

Compute the indicated products

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

Exercise 3.2 | Q 4 | Page 81

if A = [(1,2,-3),(5,0,2),(1,-1,1)], B = [(3,-1,2),(4,2,5),(2,0,3)] and C = [(4,1,2),(0,3,2),(1,-2,3)] then compute (A + B) and (B - C). Also verify that A + (B -C) = (A +B) - C

Exercise 3.2 | Q 5 | Page 81

if  A = [(2/3, 1, 5/3), (1/3, 2/3, 4/3),(7/3, 2, 2/3)] and B = [(2/5, 3/5,1),(1/5, 2/5, 4/5), (7/5,6/5, 2/5)] then compute 3A - 5B.

Exercise 3.2 | Q 6 | Page 81

Simplify  cos theta[(cos theta, sintheta),(-sin theta, cos theta)] + sin theta [(sin theta, -cos theta), (cos theta, sin theta)]

Exercise 3.2 | Q 7.1 | Page 81

Find X and Y, if X + Y = [(7,0),(2,5)] and X - Y = [(3,0),(0,3)]

Exercise 3.2 | Q 7.2 | Page 81

Find and Y, if 2X + 3Y = [(2,3),(4,0)] and 3X + 2Y = [(2, -1),(-1,5)]

Exercise 3.2 | Q 8 | Page 81

Find X, if  Y = [(3, 2),(1,4)] and 2X + Y = [(1, 0),(-3, 2)]

Exercise 3.2 | Q 9 | Page 81

Find x and y, if  2[(1,3),(0, x)]+[(y,0),(1,2)] = [(5,6),(1,8)]

Exercise 3.2 | Q 10 | Page 81

Solve the equation for x, y, z and t if 2[(x,z),(y, t)] + 3[(1,-1),(0,2)] = 3[(3,5),(4,6)]

Exercise 3.2 | Q 11 | Page 81

if x[2/3] + y[(-1),(1)] = [10/5], find values of x and y.

Exercise 3.2 | Q 12 | Page 81

Given 3[(x,y),(z,w)] = [(x,6),(-1,2W)] + [(4,x+y),(Z+W,3)] find the values of xyz and w

Exercise 3.2 | Q 13 | Page 82

If F(x) = [(cosx, -sinx,0),(sinx, cosx, 0),(0,0,1)]  show that F(x)F(y) = F(x + y)

Exercise 3.2 | Q 14.1 | Page 82

Show that [(5, -1),(6,7)][(2,1),(3,4)] != [(2,1),(3,4)][(5,-1),(6,7)]

Exercise 3.2 | Q 14.2 | Page 82

Show that [(1,2,3),(0,1,0),(1,1,0)][(-1,1,0),(0,-1,1),(2,3,4)]!=[(-1,1,0),(0,-1,1),(2,3,4)][(1,2,3),(0,1,0),(1,1,0)]

Exercise 3.2 | Q 15 | Page 82

Find A^2 - 5A + 6I if A = [(2,0,1),(2,1,3),(1,-1,0)]

Exercise 3.2 | Q 16 | Page 82

if A = [(1,0,2),(0,2,1),(2,0,3)] , prove that A^2 - 6A^2 + 7A + 2I = 0

Exercise 3.2 | Q 17 | Page 82

if A = [(3, -2),(4,-2)] and l = Matric [(1,0),(0,1)]  find k so that A^2 = kA - 2I

Exercise 3.2 | Q 18 | Page 82

if A = [(0, -tan alpha/2), (tan alpha/2, 0)] and I is the identity matrix of order 2, show that I + A = (I -A)[(cos alpha, -sin alpha),(sin alpha, cos alpha)]

Exercise 3.2 | Q 19.1 | Page 82

A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of Rs. 1,800.

Exercise 3.2 | Q 19.2 | Page 82

A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of Rs 2,000.

Exercise 3.2 | Q 20 | Page 82

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Exercise 3.2 | Q 21 | Page 83

Assume XYZW and P are matrices of order 2 x n, 3 x k, 2 x p,n x 3 and respectively. The restriction on nk and p so that PY + WY will be defined are:

A. k = 3, p = n

B. k is arbitrary, p = 2

C. p is arbitrary, k = 3

D. k = 2, p = 3

Exercise 3.2 | Q 22 | Page 83

Assume XYZW and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3, and p x k respectively. If n = p, then the order of the matrix is &X - 5Z

A p × 2 B 2 × n C n × 3 D p × n

Exercise 3.3 [Pages 88 - 90]

### NCERT solutions for Mathematics Class 12 Chapter 3 Matrices Exercise 3.3 [Pages 88 - 90]

Exercise 3.3 | Q 1 | Page 88

Find the transpose the matrices [(5),(1/2),(-1)]

Exercise 3.3 | Q 1.2 | Page 88

Find the transpose of matrices [(1,-1),(2,3)]

Exercise 3.3 | Q 1.3 | Page 88

Find the transpose of matrices [(-1,5,6),(sqrt3, 5, 6),(2,3,-1)]

Exercise 3.3 | Q 2.1 | Page 88

if A = [(-1,2,3),(5,7,9),(-2,1,1)] and B = [(-4,1,-5),(1,2,0),(1,3,1)] then verify that

(A+ B)' = A' + B'

Exercise 3.3 | Q 2.2 | Page 88

if A = [(-1,2,3),(5,7,9),(-2,1,1)] and B = [(-4,1,-5),(1,2,0),(1,3,1)] then verify that

(A- B)' = A' - B'

Exercise 3.3 | Q 3.1 | Page 88

if A' [(3,4),(-1, 2),(0,1)] and B = [((-1,2,1),(1,2,3))] then verify that (A + B)' = A' + B'

Exercise 3.3 | Q 3.2 | Page 88

if A' [(3,4),(-1, 2),(0,1)] and B = [((-1,2,1),(1,2,3))] then verify that (A - B)' = A' - B'

Exercise 3.3 | Q 4 | Page 88

if A' = [(-2,3),(1,2)] and B = [(-1,0),(1,2)]  then find (A + 2B)'

Exercise 3.3 | Q 5.1 | Page 88

For the matrices A and B, verify that (AB)′ = B'A' where

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

Exercise 3.3 | Q 5.2 | Page 88

For the matrices A and B, verify that (AB)′ = B'A'  where

A =[(0), (1),(2)] , B =[1 , 5, 7]

Exercise 3.3 | Q 6.1 | Page 89

if A = [(cos alpha, sin alpha), (-sin alpha, cos alpha)] then verify that  A'A = I

Exercise 3.3 | Q 6.2 | Page 89

if A = [(sin alpha, cos alpha),(-cos alpha, sin alpha)] then verify that  A'A = I

Exercise 3.3 | Q 7.1 | Page 89

Show that the matrix  A = [(1,-1,5),(-1,2,1),(5,1,3)] is a symmetric matrix

Exercise 3.3 | Q 7.2 | Page 89

Show that the matrix  A = [(0,1,-1),(-1,0,1),(1,-1,0)] is a skew symmetric matrix

Exercise 3.3 | Q 8.1 | Page 89

For the matrix A = [(1,5),(6,7)] verify that (A + A') is a symmetric matrix

Exercise 3.3 | Q 8.2 | Page 89

For the matrix A = [(1,5),(6,7)] verify that (A - A') is a skew symmetric matrix

Exercise 3.3 | Q 9 | Page 89

Find 1/2 (A + A') and 1/2 (A -A') When A = [(0, a, b),(-a,0,c),(-b,-c,0)]

Exercise 3.3 | Q 10.1 | Page 89

Express the matrices as the sum of a symmetric and a skew symmetric matrix:

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

Exercise 3.3 | Q 10.2 | Page 89

Express the matrices as the sum of a symmetric and a skew symmetric matrix:

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

Exercise 3.3 | Q 10.3 | Page 89

Express the matrices as the sum of a symmetric and a skew symmetric matrix:

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

Exercise 3.3 | Q 10.4 | Page 89

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

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

Exercise 3.3 | Q 11 | Page 90

If AB are symmetric matrices of same order, then AB − BA is a

A. Skew symmetric matrix B. Symmetric matrix

C. Zero matrix D. Identity matrix

Exercise 3.3 | Q 12 | Page 90

if A= [(cos alpha, -sin alpha),(sin alpha, cos alpha)] then A + A' = I if the value of α is

A pi/6

B pi/3

C pi

D (3pi)/2

Exercise 3.4 [Page 97]

### NCERT solutions for Mathematics Class 12 Chapter 3 Matrices Exercise 3.4 [Page 97]

Exercise 3.4 | Q 1 | Page 97

Find the inverse of each of the matrices, if it exists. [(1, -1),(2,3)]

Exercise 3.4 | Q 2 | Page 100

if A = [(1,1,1),(1,1,1),(1,1,1)], Prove that A" = [(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))] n in N

Exercise 3.4 | Q 2 | Page 97

Find the inverse of each of the matrices, if it exists. [(2,1),(1,1)]

Exercise 3.4 | Q 3 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 3 | Page 100

if A = [(3,-4),(1,-1)] then prove A"= [(1+2n, -4n),(n, 1-2n)] where n is any positive integer

Exercise 3.4 | Q 4 | Page 97

Find the inverse of each of the matrices, if it exists.

[(2,3),(5,7)]

Exercise 3.4 | Q 5 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 6 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 7 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 8 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 9 | Page 97

Find the inverse of each of the matrices, if it exists.

[(3,10),(2,7)]

Exercise 3.4 | Q 10 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 11 | Page 97

Find the inverse of each of the matrices, if it exists.

[(2, -6),(1, -2)]

Exercise 3.4 | Q 12 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 13 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 14 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 15 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 15 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 16 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 16 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 17 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Exercise 3.4 | Q 18 | Page 97

Matrices A and B will be inverse of each other only if

A. AB = BA

C. AB = 0, BA = I

B. AB = BA = 0

D. AB = BA = I

Exercise 3.5 [Pages 100 - 101]

### NCERT solutions for Mathematics Class 12 Chapter 3 Matrices Exercise 3.5 [Pages 100 - 101]

Exercise 3.5 | Q 1 | Page 100

Let A = [(0,1),(0,0)]show that (aI+bA)n  = anI + nan-1 bA , where I is the identity matrix of order 2 and n ∈ N

Exercise 3.5 | Q 4 | Page 100

If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix

Exercise 3.5 | Q 5 | Page 100

Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Exercise 3.5 | Q 6 | Page 100

Find the values of xyz if the matrix A = [(0,2y,z),(x,y,-z),(x , -y,z)] satisfy the equation

A'A = I

Exercise 3.5 | Q 7 | Page 100

For what values of x, [1,2,1] [(1,2,0),(2,0,1),(1,0,2)][(0),(2),(x)] = O?

Exercise 3.5 | Q 8 | Page 100

if A = [(3,1),(-1,2)]  show that  A^2 - 5A + 7I = O

Exercise 3.5 | Q 9 | Page 100

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

Exercise 3.5 | Q 10 | Page 101

A manufacturer produces three products xyz which he sells in two markets.

Annual sales are indicated below:

 Market Products I 10000 2000 18000 II 6000 20000 8000

(a) If unit sale prices of xy and are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.

Exercise 3.5 | Q 11 | Page 101

Find the matrix X so that  X[(1,2,3),(4,5,6)]= [(-7,-8,-9),(2,4,6)]

Exercise 3.5 | Q 12 | Page 101

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N

Exercise 3.5 | Q 13 | Page 101

Choose the correct answer in the following questions:

if A = [(alpha, beta),(gamma, -alpha)]` is such that A2 = I then

(A) 1 + α² + βγ = 0

(B) 1 – α² + βγ = 0

(C) 1 – α² – βγ = 0

(D) 1 + α² – βγ = 0

Exercise 3.5 | Q 14 | Page 101

If the matrix A is both symmetric and skew symmetric, then

A. A is a diagonal matrix

B. A is a zero matrix

C. A is a square matrix

D. None of these

Exercise 3.5 | Q 15 | Page 101

If A is a square matrix such that A2 = A, then (I + A)3 – 7 A is equal to:

• A

• I – A

• I

• 3A

## Solutions for Chapter 3: Matrices

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5

## NCERT solutions for Mathematics Class 12 chapter 3 - Matrices

Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Mathematics Class 12 CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT solutions for Mathematics Mathematics Class 12 CBSE, Karnataka Board PUC 3 (Matrices) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.

Concepts covered in Mathematics Class 12 chapter 3 Matrices are Introduction of Operations on Matrices, Inverse of a Matrix by Elementary Transformation, Multiplication of Two Matrices, Negative of Matrix, Properties of Matrix Addition, Transpose of a Matrix, Subtraction of Matrices, Addition of Matrices, Symmetric and Skew Symmetric Matrices, Types of Matrices, Proof of the Uniqueness of Inverse, Invertible Matrices, Multiplication of Matrices, Properties of Multiplication of Matrices, Equality of Matrices, Order of a Matrix, Matrices Notation, Introduction of Matrices, Multiplication of a Matrix by a Scalar, Properties of Scalar Multiplication of a Matrix, Properties of Transpose of the Matrices, Elementary Transformations.

Using NCERT Mathematics Class 12 solutions Matrices exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Solutions are essential questions that can be asked in the final exam. Maximum CBSE, Karnataka Board PUC Mathematics Class 12 students prefer NCERT Textbook Solutions to score more in exams.

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