#### Provide solution for rd sharma maths class 12 chapter adjoint and inverse of a matrix exercise multiple choice question question 15

option (c) 16A

Given: $A=\begin{bmatrix} 2 &0 &0 \\ 0 & 2& 0\\ 0&0 &2 \end{bmatrix}$

Hint:  you must know the multiplication concept of matrices

Solution:

\begin{aligned} &\mathrm{A}^{2}=\mathrm{A} \cdot \mathrm{A}\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right]\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right] \\ &=\left[\begin{array}{lll} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right] \\ &A^{5}= A^{2} \cdot A^{2} \cdot A \\ &A^{5}=\left[\begin{array}{lll} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right]\left[\begin{array}{lll} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right] \text { A } \end{aligned}

\begin{aligned} &=\left[\begin{array}{ccc} 16 & 0 & 0 \\ 0 & 16 & 0 \\ 0 & 0 & 16 \end{array}\right] \mathrm{A} \\ &=16\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \text { A } \\ &= 16 \mathrm{I} \mathrm{A} \\ &\Rightarrow \mathrm{A}^{5}=16 \mathrm{~A} \end{aligned}

Hence, option (c) is correct