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Sum of n Terms

If D1 and D2 ...

Question

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

A

D_{1} D_{2}

is a diagonal matrix

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B

D_{1} + D_{2}

is a diagonal matrix

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C

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

is a diagonal matrix

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D

D_{1} - D_{2}

is a diagonal matrix

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Solution

The correct option is D $D_{1} - D_{2}$ is a diagonal matrix
Let $A = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{11} & 0 & 0 & 0 0 & a_{22} & 0 & 0 0 & 0 & \cdot & 0 0 & 0 & 0 & a_{n n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

$B = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} l_{11} & 0 & 0 & 0 0 & l_{22} & 0 & 0 0 & 0 & \cdot & 0 0 & 0 & 0 & l_{n n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

then, $A B = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{11} l_{11} & 0 & 0 & 0 0 & a_{22} l_{22} & 0 & 0 0 & 0 & 0 & 0 0 & 0 & 0 & a_{n n} l_{n n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

$A \pm B = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{11} \pm l_{11} & 0 & 0 a_{22} \pm l_{22} & 0 & 0 0 & 0 & \cdot & 0 0 & 0 & 0 & a_{n n} \pm l_{n n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$
So, If $D_{1}$ and $D_{2}$ are $3 \times 3$ diagonal matrices, then $D_{1} D_{2}, D_{1} + D_{2}$ & $D_{1} - D_{2}$ are also diagonal matrices.

$A^{2} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{11}^{2} & 0 & 0 & 0 0 & a_{22}^{2} & 0 & 0 0 & 0 & 0 & 0 0 & 0 & 0 & a_{n n}^{2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

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

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