Let A = 5 - 1 6 7 and B = 2 1 3 4 , verify that A ' ' = A .

Given that, A = 5 - 1 6 7 To proof : A ' ' = A Now, A = 5 - 1 6 7 . . . . . . . . . . 1 A ' = 5 6 - 1 7 A ' ' = 5 - 1 6 7 . . . . . . . . . . 2 From equation 1 and 2 , we get A = A ' ' .

Let A = 5 - 1 6 7 and B = 2 1 3 4 , verify that 3 A ' = 3 A ' .

Given that, A = 5 - 1 6 7 To proof : 3 A ' = 3 A ' Now, A = 5 - 1 6 7 3 A = 3 5 - 1 6 7 3 A = 15 - 3 18 21 3 A ' = 15 18 - 3 21 . . . . . . . . . . . 1 A ' = 5 6 - 1 7 3 A ' = 3 5 6 - 1 7 3 A ' = 15 18 - 3 21 . . . . . . . . . . . . 2 From equation 1 and 2 , we get 3 A ' = 3 A ' .

Let A = 5 - 1 6 7 and B = 2 1 3 4 , verify that A - B ' = A ' - B ' .

Given that, A = 5 - 1 6 7 and B = 2 1 3 4 To find : A - B ' = A ' - B ' Now, A - B = 5 - 1 6 7 - 2 1 3 4 A - B = 3 - 2 3 3 A - B ' = 3 3 - 2 3 . . . . . . . . . . . . 1 A ' - B ' = 5 6 - 1 7 - 2 3 1 4 A ' - B ' = 5 - 2 6 - 3 - 1 - 1 7 - 4 A ' - B ' = 3 3 - 2 3 . . . . . . . . . . . . . 2 From equation 1 and 2 , we get A - B ' = A ' - B ' .

Let A = 5 - 1 6 7 and B = 2 1 3 4 , verify that AB ' = B ' A ' .

Given that, A = 5 - 1 6 7 and B = 2 1 3 4 To proof : AB ' = B ' A ' Now, AB = 5 - 1 6 7 2 1 3 4 AB = 5 × 2 + - 1 × 3 5 × 1 + - 1 × 4 6 × 2 + 7 × 3 6 × 1 + 7 × 4 AB = 7 1 33 34 AB ' = 7 33 1 34 . . . . . . . . . . . . 1 B ' = 2 3 1 4 A ' = 5 6 - 1 7 B ' A ' = 2 3 1 4 5 6 - 1 7 B ' A ' = 2 × 5 + 3 × - 1 2 × 6 + 3 × 7 1 × 5 + 4 × - 1 1 × 6 + 4 × 7 B ' A ' = 7 33 1 34 . . . . . . . . . . . 2 Hence, from equation 1 and 2 , we get AB ' = B ' A ' .

If A = 2 3 2 4 2 0 and B = 2 - 1 2 1 2 4 , then verify that A ' ' = A .

Given that, A = 2 3 2 4 2 0 To proof : A ' ' = A Now, A = 2 3 2 4 2 0 A ' = 2 4 3 2 2 0 A ' ' = 2 3 2 4 2 0 = A Hence, A ' ' = A .

If A = 2 3 2 4 2 0 and B = 2 - 1 2 1 2 4 , then verify that A + B ' = A ' + B '

Given that, A = 2 3 2 4 2 0 and B = 2 - 1 2 1 2 4 To proof : A + B ' = A ' + B ' Now, A + B = 2 3 2 4 2 0 + 2 - 1 2 1 2 4 A + B = 2 + 2 3 + - 1 2 + 2 4 + 1 2 + 2 0 + 4 A + B = 4 3 - 1 4 5 4 4 A + B ' = 4 5 3 - 1 4 4 4 . . . . . . . . . . . . 1 A ' = 2 4 3 2 2 0 B ' = 2 1 - 1 2 2 4 A ' + B ' = 2 4 3 2 2 0 + 2 1 - 1 2 2 4 A ' + B ' = 4 5 3 - 1 4 4 4 . . . . . . . . . . . . . . . 2 From equation 1 and 2 Hence, A + B ' = A ' + B ' .

If A = 2 3 2 4 2 0 and B = 2 - 1 2 1 2 4 , then verify that kB ' = kB '

Given that, A = 2 3 2 4 2 0 and B = 2 - 1 2 1 2 4 To proof : kB ' = kB ' Now, kB = k 2 - 1 2 1 2 4 kB = 2 k - k 2 k k 2 k 4 k kB ' = 2 k k - k 2 k 2 k 4 k . . . . . . . . . . . . . 1 B ' = 2 1 - 1 2 2 4 kB ' = k 2 1 - 1 2 2 4 kB ' = 2 k k - k 2 k 2 k 4 k . . . . . . . . . . . . . . . . . 2 From equation 1 and 2 , Hence, kB ' = kB ' .

If A ' = - 2 3 1 2 and B = - 1 0 1 2 , then find A + 2 B ' .

Given that, A ' = - 2 3 1 2 and B = - 1 0 1 2 To find : A + 2 B ' Now, A = - 2 1 3 2 [Since, A ' ' = A ] A + 2 B = - 2 1 3 2 + 2 - 1 0 1 2 A + 2 B = - 2 1 3 2 + - 2 0 2 4 A + 2 B = - 2 - 2 1 + 0 3 + 2 2 + 4 A + 2 B = - 4 1 5 6 A + 2 B ' = - 4 5 1 6 .

E M B I B E

Understanding ISC Mathematics Class 12 Volume 1>Chapter 3 - Matrices>EXERCISE 3.4

EASY

12th ICSE

IMPORTANT

Earn 100

If $A, B$ are skew-symmetric matrices of same order and $AB$ is symmetric, then show that $AB = BA$ .

Important Questions on Matrices

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 3 - Matrices>EXERCISE 3.4

Let $A = [\begin{matrix} 5 & - 1 \\ 6 & 7 \end{matrix}]$ and $B = [\begin{matrix} 2 & 1 \\ 3 & 4 \end{matrix}]$ , verify that

$(A')' = A$ .

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 3 - Matrices>EXERCISE 3.4

Let $A = [\begin{matrix} 5 & - 1 \\ 6 & 7 \end{matrix}]$ and $B = [\begin{matrix} 2 & 1 \\ 3 & 4 \end{matrix}]$ , verify that

$(3 A)' = 3 A'$ .

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 3 - Matrices>EXERCISE 3.4

Let $A = [\begin{matrix} 5 & - 1 \\ 6 & 7 \end{matrix}]$ and $B = [\begin{matrix} 2 & 1 \\ 3 & 4 \end{matrix}]$ , verify that

$(A - B)' = A' - B'$ .

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 3 - Matrices>EXERCISE 3.4

Let $A = [\begin{matrix} 5 & - 1 \\ 6 & 7 \end{matrix}]$ and $B = [\begin{matrix} 2 & 1 \\ 3 & 4 \end{matrix}]$ , verify that

$(AB)' = B' A'$ .

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 3 - Matrices>EXERCISE 3.4

If $A = [\begin{matrix} 2 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{matrix}]$ and $B = [\begin{matrix} 2 & - 1 & 2 \\ 1 & 2 & 4 \end{matrix}]$ , then verify that

${(A')}^{'} = A$ .

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 3 - Matrices>EXERCISE 3.4

If $A = [\begin{matrix} 2 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{matrix}]$ and $B = [\begin{matrix} 2 & - 1 & 2 \\ 1 & 2 & 4 \end{matrix}]$ , then verify that

$(A + B)' = A' + B'$

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 3 - Matrices>EXERCISE 3.4

If $A = [\begin{matrix} 2 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{matrix}]$ and $B = [\begin{matrix} 2 & - 1 & 2 \\ 1 & 2 & 4 \end{matrix}]$ , then verify that