Chapter 9 - Matrices Exercise Ex. 9(D)

Question 1

Find x and y, if:

Solution 1

Comparing the corresponding elements, we get,

6x - 10 = 8

6x = 18

x = 3

-2x + 14 = 4y

4y = -6+ 14 = 8

y = 2

Question 2

Find x and y, if:

Solution 2

Comparing the corresponding elements, we get,

3x + 18 = 15

3x = -3

x = -1

12x + 77 = 10y

10y = -12 + 77 = 65

y = 6.5

Question 3

If ; find x and y, if:

(i) x, y Î W (whole numbers)

(ii) x, y Î Z (integers)

Solution 3

(i) x, y Î W (whole numbers)

It can be observed that the above two equations are satisfied when x = 3 and y = 4.

 

(ii) x, y Î Z (integers)

It can be observed that the above two equations are satisfied when x = 3 and y = 4.

Question 4

 

 

 

Solution 4

(i)

 

 

 

(ii)

 

 

 

 

Question 5

Evaluate:

Solution 5

Question 6

If and 3A x M = 2B; find matrix M.

Solution 6

Let the order of matrix M be a x b.

 

3A x M = 2B

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

Comparing the corresponding elements, we get,

-3y = -10

 y =           

12x - 9y = 12

 

 

Question 7

If , find the values of a, b and c.

Solution 7

 

Comparing the corresponding elements, we get,

                                 

a + 1 = 5  a = 4

2 + b = 0  b = -2

-1 - c = 3  c = -4

Question 8

If A = ; find:

(i) A (BA)

(ii) (AB). B

Solution 8

(i)

(ii)

Question 9

Find x and y, if: 

Solution 9

Comparing the corresponding elements, we get,

5x = 5x = 1

6y = 12 y = 2

Question 10

If matrix X = and 2X - 3Y =; find the matrix 'X' and 'Y'.

Solution 10

Question 11

Given ; find the matrix X such that:

A + X = 2B + C

Solution 11

Given, A + X = 2B + C

Question 12

Find the value of x, given that A² = B,

Solution 12

 

Given, A² = B

Comparing the corresponding elements, we get,

x = 36

Question 13

If , and I is identity matrix of the same order and A^t is the transpose of matrix A, find A^t .B + BI

Solution 13

Question 14

 

Solution 14

Question 15

Let. Find A² - A + BC.

Solution 15

Question 16

Let A =. Find A² + AB + B².

Solution 16

A = 

A² = A  A = 

= 

AB = A  B = 

=

= 

B² = B x B = 

= 

= 

 A² + AB + B² = 

= 

Question 17

If and 3A - 2C = 6B, find the values of a, b and c.

Solution 17

 

Comparing the corresponding elements, we get,

3a - 8 = 24  3a = 32  a = 

24 - 2b = 0  2b = 24  b = 12

11 = 6c  c = 

Question 18

Given A =.

Find the values of p and q.

Solution 18

A = 

BA = 

C² = 

BA = C² =

By comparing,

-2q = -8  q = 4

And p = 8

Question 19

Given A = . Find AB + 2C - 4D.

Solution 19

AB = 

Question 20

Evaluate:

Solution 20

= 

= 

Question 21

 

 

Solution 21

 

 

 

 

 

 

 

 

Question 22

Solution 22

A² = 9A + MI

⇒ A² - 9A = mI ….(1)

Now, A² = AA

Substituting A² in (1), we have

A² - 9A = mI

 

Question 23

(i) Write the order of matrix X.

(ii) Find the matrix 'X'

Solution 23

 

(i) Let the order of matrix X = m × n

Order of matrix A = 2 × 2

Order of matrix B = 2 × 1

Now, AX = B

  

∴ m = 2 and n = 1

 

Thus, order of matrix X = m × n = 2 × 1

 

Multiplying (1) by 2, we get

4x + 2y = 8 ….(3)

Subtracting (2) from (3), we get

3x = 3

⇒ x = 1

Substituting the value of x in (1), we get

2(1) + y = 4

⇒ 2 + y = 4

⇒ y = 2

Question 24

Find the matrix C where C is a 2 by 2 matrix. 

Solution 24

Given: A² - 5B² = 5C

Question 25

Given matrix . Find the matrix X if, X = B² - 4B. Hence, solve for a and b given .

Solution 25

To find: a and b