Prove the following

Question:

If $\mathrm{P}=\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$ and $\mathrm{Q}=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right]$, prove that $\mathrm{PQ}=\left[\begin{array}{ccc}x a & 0 & 0 \\ 0 & y b & 0 \\ 0 & 0 & z c\end{array}\right]=\mathrm{QP}$

Solution:

Given, $P=\left[\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$ and $Q=\left[\begin{array}{ccc}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right]$

It's seen that both $\mathrm{P}$ and $\mathrm{Q}$ are diagonal matrices.

We know that, for diagonal matrices elements of product matrix are obtained by multiplying elements of matrices in the principal diagonal.

Hence,

$P Q=\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right]$

$=\left[\begin{array}{ccc}x a & 0 & 0 \\ 0 & y b & 0 \\ 0 & 0 & z c\end{array}\right]=\left[\begin{array}{ccc}a x & 0 & 0 \\ 0 & b y & 0 \\ 0 & 0 & z c\end{array}\right]=Q P$

Therefore, $\mathrm{PQ}=\mathrm{QP}$