For the following matrices verify the distributivity of matrix, m… - Vidyadip

Question

For the following matrices verify the distributivity of matrix, multiplication over matrix addtion i.e., A(B + C) = AB + AC.
$\text{A}=\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix},\text{B}=\begin{bmatrix}0&1\\1&1\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&-1\\0&1\end{bmatrix}$

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Answer

$\text{A}(\text{B}+\text{C})=\text{AB}+\text{AC}$
$\Rightarrow\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix}\begin{pmatrix}\begin{bmatrix}0&1\\1&1\end{bmatrix}+\begin{bmatrix}1&-1\\0&1\end{bmatrix}\end{pmatrix}$ $=\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix}\begin{bmatrix}0&1\\1&1\end{bmatrix}+\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix}\begin{bmatrix}1&-1\\0&1\end{bmatrix}$
$\Rightarrow\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix}\begin{bmatrix}0+1&1-1\\1+0&1+1\end{bmatrix}$ $=\begin{bmatrix}0-1&2-1\\0+1&1+1\\0+2&-1+2\end{bmatrix}+\begin{bmatrix}2-0&-2-1\\1+0&-1+1\\-1+0&1+2\end{bmatrix}$
$\Rightarrow\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix}\begin{bmatrix}1&0\\1&2\end{bmatrix}=\begin{bmatrix}-1&1\\1&2\\2&1\end{bmatrix}+\begin{bmatrix}2&-3\\1&0\\-1&3\end{bmatrix}$
$\Rightarrow\begin{bmatrix}2-1&0-2\\1+1&0+2\\-1+2&0+4\end{bmatrix}=\begin{bmatrix}-1+2&1-3\\1+1&2+0\\2-1&1+3\end{bmatrix}$
$\Rightarrow\begin{bmatrix}1&-2\\2&2\\1&4\end{bmatrix}=\begin{bmatrix}1&-2\\2&2\\1&4\end{bmatrix}$
$\therefore\ \text{LHS}=\text{RHS}$
Hence proved.

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