# NCERT Solutions for Class 12 Maths Maths Part-1 Chapter 3

## Matrices Class 12

### Exercise 3.1 : Solutions of Questions on Page Number : 64

Q1 :

In the matrix, write:

(i) The order of the matrix (ii) The number of elements,

(iii) Write the elements a13, a21, a33, a24, a23

(i) In the given matrix, the number of rows is 3 and the number of columns is 4. Therefore, the order of the matrix is 3 ×4.

(ii) Since the order of the matrix is 3 ×4, there are 3 ×4 = 12 elements in it.

(iii) a13= 19, a21= 35, a33= - 5, a24= 12, a23=

Q2 :

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

We know that if a matrix is of the order m x n, it has mn elements. Thus, to find all the possible orders of a matrix having 24 elements, we have to find all the ordered pairs of natural numbers whose product is 24.

The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and
(6, 4)

Hence, the possible orders of a matrix having 24 elements are:

1 x 24, 24 x 1, 2 x 12, 12 x 2, 3 x 8, 8 x 3, 4 x 6, and 6 x 4

(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13.

Hence, the possible orders of a matrix having 13 elements are 1 x 13 and 13 x 1.

Q3 :

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

Q4 :

Construct a 2 ×2 matrix,, whose elements are given by:

(i)

(ii)

(iii)

Q5 :

Construct a 3 ×4 matrix, whose elements are given by

(i) (ii)

Q6 :

Find the value of x, y, and zfrom the following equation:

(i) (ii)

(iii)

Q7 :

Find the value of a, b, c, and d from the equation:

Q8 :

is a square matrix, if

(A) m < n

(B) m> n

(C) m= n

(D) None of these

Q9 :

Which of the given values of xand y make the following pair of matrices equal

(A)

(B) Not possible to find

(C)

(D)

Q10 :

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

(A) 27

(B) 18

(C) 81

(D) 512

### Exercise 3.2 : Solutions of Questions on Page Number : 80

Q1 :

Let

Find each of the following

(i) (ii) (iii)

(iv) (v)

Q2 :

Compute the following:

(i) (ii)

(iii)

(v)

Q3 :

Compute the indicated products

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Q4 :

If, and, then compute and. Also, verify that

Q5 :

If andthen compute.

Q6 :

Simplify

Q7 :

Find X and Y, if

(i) and

(ii) and

Q8 :

Find X, if and

Q9 :

Find xand y, if

Q10 :

Solve the equation for x, y, zand t if

Q11 :

If, find values of xand y.

Q12 :

Given, find the values of x, y, zand

w.

Q13 :

If, show that.

Q14 :

Show that

(i)

(ii)

Q15 :

Find if

Q16 :

If, prove that

Q17 :

If and, find k so that

Q18 :

Ifand I is the identity matrix of order 2, show that

Q19 :

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:

(a) Rs 1,800 (b) Rs 2,000

Q20 :

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.

Q21 :

Assume X, Y, Z, W and P are matrices of order, and respectively. The restriction on n, k and p so that 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

Q22 :

Assume X, Y, Z, Wand Pare matrices of order, and respectively. If n= p, then the order of the matrix is

A p ×2 B 2 × n C n ×3 D p × n

### Exercise 3.3 : Solutions of Questions on Page Number : 88

Q1 :

Find the transpose of each of the following matrices:

(i) (ii) (iii)

Q2 :

If and, then verify that

(i)

(ii)

Q3 :

If and, then verify that

(i)

(ii)

Q4 :

If and, then find

Q5 :

For the matrices Aand B, verify that (AB)“² = where

(i)

(ii)

Q6 :

If (i) , then verify that

(ii) , then verify that

Q7 :

(i) Show that the matrix is a symmetric matrix

(ii) Show that the matrix is a skew symmetric matrix

Q8 :

For the matrix, verify that

(i) is a symmetric matrix

(ii) is a skew symmetric matrix

Q9 :

Find and, when

Q10 :

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

(i)

(ii)

(iii)

(iv)

Q11 :

If A, Bare symmetric matrices of same order, then AB- BAis a

A. Skew symmetric matrix B. Symmetric matrix

C. Zero matrix D. Identity matrix

Q12 :

If, then, if the value of α is

A. B.

C. π D.

### Exercise 3.4 : Solutions of Questions on Page Number : 97

Q1 :

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

Q2 :

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

Q3 :

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

Q4 :

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

Q5 :

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

Q6 :

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

Q7 :

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

Q8 :

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

Q9 :

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

Q10 :

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

Q11 :

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

Q12 :

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

Q13 :

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

Q14 :

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

Q15 :

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

Q16 :

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

Q17 :

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

Q18 :

Matrices A and Bwill be inverse of each other only if

A. AB= BA

C. AB= 0, BA = I

B. AB= BA = 0

D. AB= BA = I

### Exercise Miscellaneous : Solutions of Questions on Page Number : 100

Q1 :

Let, show that, where I is the identity matrix of order 2 and n N

Q2 :

If, prove that

Q3 :

If, then prove where n is any positive integer

Q4 :

If Aand B are symmetric matrices, prove that AB- BAis a skew symmetric matrix.

Q5 :

Show that the matrix is symmetric or skew symmetric according as Ais symmetric or skew symmetric.

Q6 :

Find the values of x, y, zif the matrix satisfy the equation

Q7 :

For what values of?

Q8 :

If, show that

Q9 :

Find x, if

Q10 :

A manufacturer produces three products x, y, z 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 x, y and z 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.

Q11 :

Find the matrix Xso that

Q12 :

If Aand B are square matrices of the same order such that AB= BA, then prove by induction that. Further, prove that for all n N

Q13 :

Choose the correct answer in the following questions:

If is such that then

A.

B.

C.

D.

Q14 :

If the matrix Ais both symmetric and skew symmetric, then

A. Ais a diagonal matrix

B. Ais a zero matrix

C. Ais a square matrix

D. None of these

Q15 :

If Ais square matrix such that then is equal to

A. A B. I - A C. I D. 3A