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.
$\text{A}=\begin{bmatrix}1&-1\\0&2\end{bmatrix},\text{B}=\begin{bmatrix}-1&0\\2&1\end{bmatrix}$ and $\text{C}=\begin{bmatrix}0&1\\1&-1\end{bmatrix}$

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Answer

$\text{A}(\text{B}+\text{C})=\text{AB}+\text{AC}$
$\Rightarrow\begin{bmatrix}1&-1\\0&2\end{bmatrix}\begin{pmatrix}\begin{bmatrix}-1&0\\2&1\end{bmatrix}+\begin{bmatrix}0&1\\1&-1\end{bmatrix}\end{pmatrix}$ $=\begin{bmatrix}1&-1\\0&2 \end{bmatrix}\begin{bmatrix}-1&0\\2&1\end{bmatrix}+\begin{bmatrix}1&-1\\0&2\end{bmatrix}\begin{bmatrix}0&1\\1&-1\end{bmatrix}$
$\Rightarrow\begin{bmatrix}1&-1\\0&2 \end{bmatrix}\begin{bmatrix}-1+0&0+1\\2+1&1-1\end{bmatrix}$ $=\begin{bmatrix}-1-2&0-1\\0+4&0+2\end{bmatrix}+\begin{bmatrix}0-1&1+1\\0+2&0-2\end{bmatrix}$
$\Rightarrow\begin{bmatrix}1&-1\\0&2 \end{bmatrix}\begin{bmatrix}-1&1\\0&3\end{bmatrix}=\begin{bmatrix}-3&-1\\4&2\end{bmatrix}+\begin{bmatrix}-1&2\\2&-2\end{bmatrix}$
$\Rightarrow\begin{bmatrix}-1-3&1-0\\0+6&0+0\end{bmatrix}=\begin{bmatrix}-3-1&-1+2\\4+2&2-2\end{bmatrix}$
$\Rightarrow\begin{bmatrix}-4&1\\6&0\end{bmatrix}=\begin{bmatrix}-4&1\\6&0\end{bmatrix}$
$\therefore\ \text{LHS}=\text{RHS}$

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