If $D_{1}$ and $D_{2}$ are two $3 x 3$ diagonal matrices, then

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$D_{1}^{n}$ is a diagonal matrix $\forall n \in N$

$D_{1}^{2} + D_{2}^{2}$ is a diagonal matrix

$D_{1} D_{2}$ is a diagonal matrix

$D_{1} D_{2} = D_{2} D_{1}$

answer is A, B, C, D.

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Detailed Solution

Let

$D_{1} = [\begin{array}{ccc} d_{1} & 0 & 0 \\ 0 & d_{2} & 0 \\ 0 & 0 & d_{3} \end{array}] \Rightarrow D_{1}^{n} = [\begin{array}{ccc} d_{1}^{n} & 0 & 0 \\ 0 & d_{2}^{n} & 0 \\ 0 & 0 & d_{3}^{n} \end{array}] =$ diagonal matrix

and $D_{2} = [\begin{array}{ccc} g_{1} & 0 & 0 \\ 0 & g_{2} & 0 \\ 0 & 0 & g_{3} \end{array}] \Rightarrow D_{2}^{n} = [\begin{array}{ccc} g_{1}^{n} & 0 & 0 \\ 0 & g_{2}^{n} & 0 \\ 0 & 0 & g_{3}^{n} \end{array}]$ $=$ Diagonal matrix $∴ D_{1} D_{2} = [\begin{array}{ccc} d_{1} g_{1} & 0 & 0 \\ 0 & d_{2} g_{2} & 0 \\ 0 & 0 & d_{3} g_{3} \end{array}] =$ diagonal matrix

and $D_{2} D_{1} = [\begin{array}{ccc} g_{1} d_{1} & 0 & 0 \\ 0 & g_{2} d_{2} & 0 \\ 0 & 0 & g_{3} d_{3} \end{array}] = D_{1} D_{2}$

Also $D_{1}^{2} + D_{2}^{2} = [\begin{array}{ccc} d_{1}^{2} + g_{1}^{2} & 0 & 0 \\ 0 & d_{2}^{2} + g_{2}^{2} & 0 \\ 0 & 0 & d_{3}^{2} + g_{3}^{2} \end{array}]$ = Diagonal matrix.