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Transpose of a Matrix

If D1 and ...

Question

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

A

D_{1} D_{2}

is diagonal matrix

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B

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

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C

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

is a diagonal matrix

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D

none of these

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Solution

The correct options are
A $D_{1}^{2} + D_{2}^{2}$ is a diagonal matrix
C $D_{1} D_{2}$ is diagonal matrix
D $D_{1} D_{2} = D_{2} D_{1}$
Let $D_{1} = ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦, D_{2} = ⎡ ⎢ ⎣ \begin{matrix} x & 0 & 0 0 & y & 0 0 & 0 & z \end{matrix} ⎤ ⎥ ⎦$
Then
$D_{1} D_{2} = ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x & 0 & 0 0 & y & 0 0 & 0 & z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a x & 0 & 0 0 & b y & 0 0 & 0 & c z \end{matrix} ⎤ ⎥ ⎦$

$D_{2} D_{1} = ⎡ ⎢ ⎣ \begin{matrix} x & 0 & 0 0 & y & 0 0 & 0 & z \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} x a & 0 & 0 0 & y b & 0 0 & 0 & z c \end{matrix} ⎤ ⎥ ⎦$
As $a x = x a, b y = y b, c z = z c$
$D_{1} D_{2} = D_{2} D_{1}$
${D_{1}}^{2} + {D_{2}}^{2} = ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} x & 0 & 0 0 & y & 0 0 & 0 & z \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x & 0 & 0 0 & y & 0 0 & 0 & z \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎣ \begin{matrix} a^{2} & 0 & 0 0 & b^{2} & 0 0 & 0 & c^{2} \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} x^{2} & 0 & 0 0 & y^{2} & 0 0 & 0 & z^{2} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} {a^{2} + x}^{2} & 0 & 0 0 & {b^{2} + y}^{2} & 0 0 & 0 & {c^{2} + z}^{2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

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