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Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices3 Marks
Question
Find the matrix A such that
$\begin{bmatrix}2&1&3\end{bmatrix}\begin{bmatrix}-1&0&-1\\-1&1&0\\0&1&1\end{bmatrix}\begin{bmatrix}1\\0\\-1\end{bmatrix}=\text{A}$
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Answer

Let $\text{A}=[\text{x}]$
$ \Rightarrow\begin{bmatrix}2&1&3\end{bmatrix}\begin{bmatrix}-1&0&-1\\-1&1&0\\0&1&1\end{bmatrix}\begin{bmatrix}1\\0\\-1\end{bmatrix}=\text{A}$
$ \Rightarrow\begin{bmatrix}2&1&3\end{bmatrix}\begin{bmatrix}-1&0&-1\\-1&1&0\\0&1&1\end{bmatrix}\begin{bmatrix}1\\0\\-1\end{bmatrix}=[\text{x}]$
$ \Rightarrow\begin{bmatrix}-2-1+0&0+1+3&-2+0+3\end{bmatrix}\begin{bmatrix}1\\0\\-1\end{bmatrix}=[\text{x}]$
$ \Rightarrow\begin{bmatrix}-3&4&1\end{bmatrix}\begin{bmatrix}1\\0\\-1\end{bmatrix}=[\text{x}]$
$ \Rightarrow\begin{bmatrix}-3+0-1\end{bmatrix}=[\text{x}]$
$\Rightarrow\begin{bmatrix}-4\end{bmatrix}=[\text{x}]$
The corresponding elements of two equal matrices are equal.
$\therefore\ \text{x}=-4$
$\therefore\ \text{A}=[-4]$

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