Question

If `P=[[x,0,0],[0,y,0],[0,0,z]]` and `Q=[[a,0,0],[0,b,0],[0,0,c]]` prove that `PQ=[[xa,0,0],[0,yb,0],[0,0,zc]]=QP`

Answer 1

\[PQ = \begin{bmatrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{bmatrix}\begin{bmatrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{bmatrix}\]
\[ = \begin{bmatrix}xa + 0 + 0 & 0 + 0 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + yb + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + 0 + 0 & 0 + 0 + zc\end{bmatrix}\]
\[ = \begin{bmatrix}xa & 0 & 0 \\ 0 & yb & 0 \\ 0 & 0 & zc\end{bmatrix} . . . (4)\]\[QP = \begin{bmatrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{bmatrix}\begin{bmatrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{bmatrix}\]
\[ = \begin{bmatrix}ax + 0 + 0 & 0 + 0 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + by + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + 0 + 0 & 0 + 0 + cz\end{bmatrix}\]
\[ = \begin{bmatrix}xa & 0 & 0 \\ 0 & yb & 0 \\ 0 & 0 & zc\end{bmatrix} . . . (5)\]\[PQ = \begin{bmatrix}xa & 0 & 0 \\ 0 & yb & 0 \\ 0 & 0 & zc\end{bmatrix} = QP\]