For the following matrices A and B , verify that ( A B ) ' = B ' A ' A = 1 2 3 4 , B = 1 5

Given: A = 1 2 3 4 , B = 1 5 A ' = 1 3 2 4 , B ' = 1 5 Now, A B = 1 2 3 4 × 1 5 = 1 + 10 3 + 20 = 11 23 A B ' = 11 23 B ' A ' = 1 5 1 3 2 4 = 1 + 10 3 + 20 = 11 23 Hence, verified that A B ' = B ' A ' .

For the following matrices A and B , verify that ( A B ) ' = B ' A ' A = 1 2 3 4 , B = 1 5

Given: A = 1 2 3 4 , B = 1 5 A ' = 1 3 2 4 , B ' = 1 5 Now, A B = 1 2 3 4 × 1 5 = 1 + 10 3 + 20 = 11 23 A B ' = 11 23 B ' A ' = 1 5 1 3 2 4 = 1 + 10 3 + 20 = 11 23 Hence, verified that A B ' = B ' A ' .

Show that 2 - 1 3 - 5 3 1 - 3 2 3 is inverse of the matrix - 7 - 9 10 - 12 - 15 17 1 1 - 1 .

A pair of matrices are inverse of each other if their product is equal to unit matrix. That is, A B = I ⇒ A = B - 1 & B = A - 1 . 2 - 1 3 - 5 3 1 - 3 2 3 × - 7 - 9 10 - 12 - 15 17 1 1 - 1 = - 14 + 12 + 3 - 18 + 15 + 3 20 - 17 - 3 35 - 36 + 1 45 - 45 + 1 - 50 + 51 - 1 21 - 24 + 3 27 - 30 + 3 - 30 + 34 - 3 = 1 0 0 0 1 0 0 0 1 = I Therefore, the given matrices are inverse of each other.

Prove that the matrix cos x - sin x 0 sin x cos x 0 0 0 1 and cos x sin x 0 - sin x cos x 0 0 0 1 are inverse of each other

A pair of matrices are inverse of each other if their product is equal to unit matrix. That is, A B = I ⇒ A = B - 1 & B = A - 1 . cos x - sin x 0 sin x cos x 0 0 0 1 × cos x sin x 0 - sin x cos x 0 0 0 1 cos 2 x + sin 2 x cos x sin x - sin x cos x 0 cos x sin x - sin x cos x cos 2 x + sin 2 x 0 0 0 1 = 1 0 0 0 1 0 0 0 1 = I Therefore, the given matrices are inverse of each other.

Express 2 4 - 1 3 5 8 1 - 2 1 as the sum of the symmetric and a skew-symmetric matrix.

According to the property of symmetric and skew symmetric matrices, a square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix. Given: 2 4 - 1 3 5 8 1 - 2 1 Let A = 2 4 - 1 3 5 8 1 - 2 1 ⇒ A T = 2 3 1 4 5 - 2 - 1 8 1 1 2 A + A T = 1 2 2 4 - 1 3 5 8 1 - 2 1 + 2 3 1 4 5 - 2 - 1 8 1 = 1 2 4 7 0 7 10 6 0 6 2 ⇒ 1 2 A + A T T = 1 2 4 7 0 7 10 6 0 6 2 which implies 1 2 A + A T is symmetric matrix. 1 2 A - A T = 1 2 2 4 - 1 3 5 8 1 - 2 1 - 2 3 1 4 5 - 2 - 1 8 1 = 1 2 0 1 - 2 - 1 0 10 2 - 10 0 ⇒ 1 2 A - A T T = 1 2 0 1 - 2 - 1 0 10 2 - 10 0 T = 1 2 0 - 1 2 1 0 - 10 - 2 10 0 ⇒ 1 2 A - A T T = - 1 2 0 1 - 2 - 1 0 10 2 - 10 0 = - 1 2 A - A T , which implies 1 2 A - A T is skew symmetric matrix. Hence, 2 4 - 1 3 5 8 1 - 2 1 = 1 2 4 7 0 7 10 6 0 6 2 + 1 2 0 - 1 2 1 0 - 10 - 2 10 0

Find inverse of A = 1 2 4 5 by elementary row transformation method

Let A = I × A 1 2 4 5 = 1 0 0 1 A 1 2 0 - 3 = 1 0 - 4 1 A , applying R 2 → R 2 - 4 R 1 1 2 0 1 = 1 0 4 3 - 1 3 A , applying R 2 → 1 - 3 R 2 1 0 0 1 = - 5 3 2 3 4 3 - 1 3 A , applying R 1 → R 1 - 2 R 2 Therefore, - 5 3 2 3 4 3 - 1 3 A = I ⇒ A - 1 = - 5 3 2 3 4 3 - 1 3

Find the values of a , b , c and d from the following equations: 2 a + b a - 2 b 5 c - d 4 c + 3 d = 4 - 3 11 24 .

Given that, 2 a + b a - 2 b 5 c - d 4 c + 3 d = 4 - 3 11 24 To find : value of a , b , c and d Now, 2 a + b a - 2 b 5 c - d 4 c + 3 d = 4 - 3 11 24 Equating the corresponding elements... 2 a + b = 4 ⇒ b = 4 - 2 a . . . . . . . . . . . 1 a - 2 b = - 3 . . . . . . . . . . . . . 2 ⇒ a - 2 4 - 2 a = - 3 [from (1)] ⇒ a - 8 + 4 a = - 3 ⇒ a = 1 Put the value of a in equation 1 b = 4 - 2 1 ⇒ b = 2 5 c - d = 11 ⇒ 5 c - 11 = d . . . . . . . . . . 3 4 c + 3 d = 24 . . . . . . . . . . . 4 ⇒ 4 c + 3 5 c - 11 = 24 [from (3)] ⇒ c = 3 Put the value of c in 3 5 × 3 - 11 = d ⇒ d = 4 . Hence, a = 1 , b = 2 , c = 3 , d = 4 .

If A = 8 0 4 - 2 3 6 and B = 2 - 2 4 2 - 5 1 , then find the matrix X , such that 2 A + 3 X = 5 B .

A = 8 0 4 - 2 3 6 and B = 2 - 2 4 2 - 5 1 Since, 2 A + 3 X = 5 B ⇒ 2 8 0 4 - 2 3 6 + 3 X = 5 2 - 2 4 2 - 5 1 ⇒ 16 0 8 - 4 6 12 + 3 X = 10 - 10 20 10 - 25 5 ⇒ 3 X = 10 - 10 20 10 - 25 5 - 16 0 8 - 4 6 12 ⇒ X = 1 3 - 6 - 10 12 14 - 31 - 7

E M B I B E

Gamma Question Bank for Engineering Mathematics>Chapter 19 - Matrices>Assignment>Q 12

EASY

JEE Main/Advance

IMPORTANT

Earn 100

For the following matrices $A$ and $B$ , verify that $(A B)^{'} = B^{'} A^{'}$

$A = [\begin{array}{l} 1 & 2 \\ 3 & 4 \end{array}], B = [\begin{array}{l} 1 \\ 5 \end{array}]$

Important Questions on Matrices

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 19 - Matrices>Assignment>Q 13

Show that $[\begin{array}{r} 2 & - 1 & 3 \\ - 5 & 3 & 1 \\ - 3 & 2 & 3 \end{array}]$ is inverse of the matrix $[\begin{array}{r} - 7 & - 9 & 10 \\ - 12 & - 15 & 17 \\ 1 & 1 & - 1 \end{array}]$ .