Let A = x 1 2 - 1 y 3 - 2 - 3 z is skew-symmetric matrix, then find x + y + z

A matrix is called a skew symmetric matrix if it is negative of its transpose. Given: A = x 1 2 - 1 y 3 - 2 - 3 z ⇒ A = - A T ⇒ x 1 2 - 1 y 3 - 2 - 3 z = - x - 1 - 2 1 y - 3 2 3 z ⇒ x 1 2 - 1 y 3 - 2 - 3 z = - x 1 2 - 1 - y 3 - 2 - 3 - z Equating the corresponding elements we get, x = - x ⇒ 2 x = 0 ⇒ x = 0 y = - y ⇒ 2 y = 0 ⇒ y = 0 z = - z ⇒ 2 z = 0 ⇒ z = 0 ∴ x + y + z = 0 Aliter: Since the diagonal elements of a skew symmetric matrix is 0 , hence the value of each of x , y & z is 0 . ∴ x + y + z = 0

Let A = x 1 2 - 1 y 3 - 2 - 3 z is skew-symmetric matrix, then find x + y + z

A matrix is called a skew symmetric matrix if it is negative of its transpose. Given: A = x 1 2 - 1 y 3 - 2 - 3 z ⇒ A = - A T ⇒ x 1 2 - 1 y 3 - 2 - 3 z = - x - 1 - 2 1 y - 3 2 3 z ⇒ x 1 2 - 1 y 3 - 2 - 3 z = - x 1 2 - 1 - y 3 - 2 - 3 - z Equating the corresponding elements we get, x = - x ⇒ 2 x = 0 ⇒ x = 0 y = - y ⇒ 2 y = 0 ⇒ y = 0 z = - z ⇒ 2 z = 0 ⇒ z = 0 ∴ x + y + z = 0 Aliter: Since the diagonal elements of a skew symmetric matrix is 0 , hence the value of each of x , y & z is 0 . ∴ x + y + z = 0

If A = cos α - sin α sin α cos α , then for what value of α is A an identity matrix?

Given that, A = cos α - sin α sin α cos α To find : α is Now, A is an identity matrix So, cos α - sin α sin α cos α = 1 0 0 1 Equating the corresponding equations... cos α = 1 cos α = cos 0 ° α = 0 .

If 1 2 3 4 3 1 2 5 = 7 11 k 23 , then write the value of k .

Given that, 1 2 3 4 3 1 2 5 = 7 11 k 23 To find : k Now, Using matrix multiplication .... 1 × 3 + 2 × 2 1 × 1 + 2 × 5 3 × 3 + 4 × 2 3 × 1 + 4 × 5 = 7 11 k 23 7 11 17 23 = 7 11 k 23 Equating the corresponding elements k = 17 Hence, k = 17 .

If A = 2 0 1 2 1 3 1 - 1 0 , then find the value of A 2 - 3 A + 2 I .

A = 2 0 1 2 1 3 1 - 1 0 ⇒ A 2 = 2 0 1 2 1 3 1 - 1 0 × 2 0 1 2 1 3 1 - 1 0 = 4 + 0 + 1 0 + 0 - 1 2 + 0 + 0 4 + 2 + 3 0 + 1 - 3 2 + 3 + 0 2 - 2 + 0 0 - 1 + 0 1 - 3 + 0 = 5 - 1 2 9 - 2 5 0 - 1 - 2 3 A = 3 2 0 1 2 1 3 1 - 1 0 = 6 0 3 6 3 9 3 - 3 0 2 I = 2 1 0 0 0 1 0 0 0 1 = 2 0 0 0 2 0 0 0 2 Putting in the matrix polynomial we get, A 2 - 3 A + 2 I = 5 - 1 2 9 - 2 5 0 - 1 - 2 - 6 0 3 6 3 9 3 - 3 0 + 2 0 0 0 2 0 0 0 2 = 5 - 6 + 2 - 1 + 0 + 0 2 - 3 + 0 9 - 6 + 0 - 2 - 3 + 2 5 - 9 + 0 0 - 3 + 0 - 1 + 3 + 0 - 2 + 0 + 2 = 1 - 1 - 1 3 - 3 - 4 - 3 2 0

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.

E M B I B E

Gamma Question Bank for Engineering Mathematics>Chapter 19 - Matrices>Assignment>Q 8

EASY

JEE Main/Advance

IMPORTANT

Earn 100

Let $A = [\begin{array}{r} x & 1 & 2 \\ - 1 & y & 3 \\ - 2 & - 3 & z \end{array}]$ is skew-symmetric matrix, then find $x + y + z$

Important Questions on Matrices

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 19 - Matrices>Assignment>Q 9

If

A = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}]

, then for what value of

x, y & z

is

A

an identity matrix?

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 19 - Matrices>Assignment>Q 10

If

[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} 3 & 1 \\ 2 & 5 \end{matrix}] = [\begin{matrix} 7 & 11 \\ k & 23 \end{matrix}]

, then write the value of

k

.

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 19 - Matrices>Assignment>Q 11

If

A = [\begin{array}{r} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & - 1 & 0 \end{array}]

, then find the value of

A^{2} - 3 A + 2 I

.

EASY

JEE Main/Advance

IMPORTANT

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

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}]

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 $[\begin{array}{r} - 7 & - 9 & 10 \\ - 12 & - 15 & 17 \\ 1 & 1 & - 1 \end{array}]$ .