Chapter 9 - Matrices Exercise Ex. 9(C)

Question 1

Evaluate: if possible:

Solution 1

The number of columns in the first matrix is not equal to the number of rows in the second matrix. Thus, the product is not possible.

Question 2

If  and I is a unit matrix of order 2  2, find:

(i) AB (ii) BA (iii) AI

(iv) IB (v) A² (vi) B²A

Solution 2

Question 3

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Solution 3

Question 4

Find x and y, if:

Solution 4

Comparing the corresponding elements, we get,

5x - 2 = 8  x = 2

20 + 3x = y  y = 20 + 6 = 26

Comparing the corresponding elements, we get,

x = 2

-3 + y = -2  y = 1

Question 5

If  , find:

(i) (AB)C (ii) A(BC)

Is A(BC) = (AB)C?

Solution 5

Hence, A(BC) = (AB)C

Question 6

Given  , find; if possible:

(i) AB (ii) BA (iii)A²

Solution 6

(iii) Product AA (=A²) is not possible as the number of columns of matrix A is not equal to its number of rows.

Question 7

Solution 7

Question 8

If M =  and I is a unit matrix of the same order as that of M; show that:

M² = 2M + 3I

Solution 8

 

Hence, M² = 2M + 3I.

Question 9

If  and BA= M², find the values of a and b.

Solution 9

 

Given, BA = M²

Comparing the corresponding elements, we get,

a = 2 and  -2b = -2 b = 1

Question 10

Given , find:

(i) A - B (ii) A²

(iii) AB (iv) A² - AB + 2B

Solution 10

Question 11

If ; find:

(i) (A + B)² (ii) A² + B²

(iii) Is (A + B)² = A² + B²?

Solution 11

(iii) Clearly, (A + B)²  A² + B²

Question 12

Find the matrix A, if B = and B² = B + A.

Solution 12

B² = B + A

A = B² - B

A = 2(B² - B)

Question 13

If A =  and A² = I; find a and b.

Solution 13

It is given that A² = I.

 

Comparing the corresponding elements, we get,

1 + a = 1

Therefore, a = 0

 

 

 

-1 + b = 0 

Therfore, b = 1

Question 14

If ; then show that:

(i) A (B + C) = AB + AC

(ii) (B - A)C = BC - AC.

Solution 14

Question 15

If , simplify:

A² + BC.

Solution 15

Question 16(i)

Solve for x and y:

Solution 16(i)

Question 16(ii)

Solve for x and y:

Solution 16(ii)

Question 16(iii)

Solution 16(iii)

Question 17

In each case given below, find:

(a) The order of matrix M.

(b) The matrix M.

 

Solution 17

We know, the product of two matrices is defined only when the number of columns of first matrix is equal to the number of rows of the second matrix.

 

(i) Let the order of matrix M be a x b.

Clearly, the order of matrix M is 1 x 2.

 

Comparing the corresponding elements, we get,

a = 1 and a + 2b = 2  2b = 2 - 1 = 1  b = 

 

(ii) Let the order of matrix M be a x b.

Clearly, the order of matrix M is 2 x 1.

 

Comparing the corresponding elements, we get,

a + 4b = 13 ....(1)

2a + b = 5 ....(2)

 

Multiplying (2) by 4, we get,

8a + 4b = 20 ....(3)

 

Subtracting (1) from (3), we get,

7a = 7  a = 1

From (2), we get,

b = 5 - 2a = 5 - 2 = 3

Question 18

If ; find the value of x, given that: A² = B.

Solution 18

Question 19

Solution 19

Question 20

If A and B are any two 2 x 2 matrices such that AB = BA = B and B is not a zero matrix, what can you say about the matrix A?

Solution 20

AB = BA = B

We know that I × B = B × I = B, where I is the identity matrix.

Hence, A is an identity matrix.

Question 21

Given  and that AB = A + B; find the values of a, b and c.

Solution 21

Comparing the corresponding elements, we get,

3a = 3 + a

2a = 3 

a =            

 

 

3b = b  b = 0

4c = 4 + c  3c = 4  c = 

Question 22

If , then compute:

(i) P² - Q² (ii) (P + Q) (P - Q)

Is (P + Q) (P - Q) = P² - Q² true for matrix algebra?

Solution 22

Clearly, it can be said that:

(P + Q) (P - Q) = P² - Q² not true for matrix algebra.

Question 23

Given the matrices:

. Find:

(i) ABC (ii) ACB.

State whether ABC = ACB.

Solution 23

 

 

Hence, ABC ≠ ACB.

Solution 23

 

 

Hence, ABC ≠ ACB.

Question 24

If ; find each of the following and state if they are equal:

(i) CA + B (ii) A + CB

Solution 24

 

Thus, CA + B   A + CB

Question 25

If ; find the matrix X such that AX = B.

Solution 25

Clearly, the order of matrix X is 2 x 1.

 

Comparing the two matrices, we get,

2x + y = 3 … (1)

x + 3y = -11 … (2)

Multiplying (1) with 3, we get,

6x + 3y = 9 … (3)

Subtracting (2) from (3), we get,

5x = 20

x = 4

From (1), we have:

y = 3 - 2x = 3 - 8 = -5

 

Question 26

If, find (A - 2I) (A - 3I).

Solution 26

Question 27

If , find:

(i) A^t. A (ii) A. A^t

Where A^t is the transpose of matrix A.

Solution 27

Question 28

If, show that: 6M - M² = 9I; where I is a 2 x 2 unit matrix.

Solution 28

Hence, proved.

Question 29

If; find x and y such that PQ = null matrix.

Solution 29

 

Comparing the corresponding elements, we get,

2x + 12 = 0

thus, x = -6

6 + 6y = 0

thus, y = -1

Question 30

Evaluate without using tables:

Solution 30

Question 31

State, with reason, whether the following are true or false. A, B and C are matrices of order 2 x 2.

(i) A + B = B + A

(ii) A - B = B - A

(iii) (B. C). A = B. (C. A)

(iv) (A + B). C = A. C + B. C

(v) A. (B - C) = A. B - A. C

(vi) (A - B). C = A. C - B. C

(vii) A² - B² = (A + B) (A - B)

(viii) (A - B)² = A² - 2A. B + B²

Solution 31

(i) True.

Addition of matrices is commutative.

(ii) False.

Subtraction of matrices is not commutative.

(iii) True.

Multiplication of matrices is associative.

(iv) True.

Multiplication of matrices is distributive over addition.

(v) True.

Multiplication of matrices is distributive over subtraction.

(vi) True.

Multiplication of matrices is distributive over subtraction.

(vii) False.

Laws of algebra for factorization and expansion are not applicable to matrices.

(viii) False.

Laws of algebra for factorization and expansion are not applicable to matrices.