If $A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$, then $A^4$ is equal to:
(a) $\begin{bmatrix} 16 & 0 \\ 0 & 81 \end{bmatrix}$
(b) $\begin{bmatrix} 8 & 0 \\ 0 & 12 \end{bmatrix}$
(c) $\begin{bmatrix} 16 & 0 \\ 0 & 12 \end{bmatrix}$
(d) $\begin{bmatrix} 8 & 0 \\ 0 & 81 \end{bmatrix}$
Step-by-Step Solution
For diagonal matrix, A^4 = [[2^4, 0],[0, 3^4]] = [[16, 0],[0, 81]]. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Selecting $\begin{bmatrix} 16 & 0 \\ 0 & 81 \end{bmatrix}$: 1.0 Mark
Correct Answer: $\begin{bmatrix} 16 & 0 \\ 0 & 81 \end{bmatrix}$