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Home  >>  CBSE XII  >>  Math  >>  Matrices
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If $P=\begin{bmatrix}x & 0 &0\\0 & y & 0\\0 & 0 & z\end{bmatrix}\;$ and $\;Q=\begin{bmatrix}a & 0 & 0\\0 & b & 0\\0 & 0 & c\end{bmatrix}$ prove that $PQ=\begin{bmatrix}xa & 0 & 0\\0 & yb & 0\\0 & 0 & zc\end{bmatrix}=QP$

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Toolbox:
  • If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Step1:
Given
$P=\begin{bmatrix}x & 0 & 0\\0 & y & 0\\0 & 0 &z\end{bmatrix} $
$Q=\begin{bmatrix}a & 0 & 0\\0 & b & 0\\0 & 0 &c\end{bmatrix}$
$PQ=\begin{bmatrix}x & 0 & 0\\0 & y & 0\\0 & 0 &z\end{bmatrix}\begin{bmatrix}a & 0 & 0\\0 & b & 0\\0 & 0 &c\end{bmatrix}$
$\;\;\;=\begin{bmatrix}xa+0+0 & 0+0+0 & 0+0+0\\0+0+0 & 0+yb+0 & 0+0+0\\0+0+0 & 0+0+0 &0+0+zc\end{bmatrix}$
$\;\;\;=\begin{bmatrix}xa & 0 & 0\\0 & yb & 0\\0 & 0 &zc\end{bmatrix}$
Step2:
$QP=\begin{bmatrix}a & 0 & 0\\0 & b & 0\\0 & 0 &c\end{bmatrix}\begin{bmatrix}x & 0 & 0\\0 & y & 0\\0 & 0 &z\end{bmatrix}$
$\;\;\;=\begin{bmatrix}ax+0+0 & 0+0+0 & 0+0+0\\0+0+0 & 0+by+0 & 0+0+0\\0+0+0 & 0+0+0 &0+0+cz\end{bmatrix}$
$\;\;\;=\begin{bmatrix}ax & 0 & 0\\0 & by & 0\\0 & 0 &cz\end{bmatrix}$
$\Rightarrow Hence$
$PQ=\begin{bmatrix}xa & 0 & 0\\0 & yb & 0\\0 & 0 & cz\end{bmatrix}=QP.$
answered Mar 23, 2013 by sharmaaparna1
 

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