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

D_{1} D_{2}

is a diagonal matrix

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D_{1} + D_{2}

is a diagonal matrix

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D_{1}^{2} + D_{2}^{2}

is a diagonal matrix

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1, 2, 3 are correct

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Solution

The correct option is D 1, 2, 3 are correct
If $D_{1}$ and $D_{2}$ are diagonal matrix then
$D_{1} D_{2}, D_{1} + D_{2}$ is also a diagonal matrix.
$A = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{11} a_{22} . a_{n n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

$B = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} l_{11} l_{22} . l_{n n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

then,
$A B = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{11} l_{11} a_{22} l_{22} . a_{n n} l_{n n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

$A + B = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{11} + l_{11} a_{22} + l_{22} . a_{n n} + l_{n n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

$A^{2} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{11}^{2} a_{22}^{2} . a_{n n}^{2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

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

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