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Question
Find the matrix A such that [2 1 3 ] `[[-1,0,-1],[-1,1,0],[0,1,1]] [[1],[0],[-1]]=A`
Solution
\[\left( iv \right) Let A = \begin{bmatrix}x\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}2 & 1 & 3\end{bmatrix}\begin{bmatrix}- 1 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 1\end{bmatrix}\begin{bmatrix}1 \\ 0 \\ - 1\end{bmatrix} = A\]
\[ \Rightarrow \begin{bmatrix}2 & 1 & 3\end{bmatrix}\begin{bmatrix}- 1 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 1\end{bmatrix}\begin{bmatrix}1 \\ 0 \\ - 1\end{bmatrix} = \begin{bmatrix}x\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}- 2 - 1 + 0 & 0 + 1 + 3 & - 2 + 0 + 3\end{bmatrix}\begin{bmatrix}1 \\ 0 \\ - 1\end{bmatrix} = \begin{bmatrix}x\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}- 3 & 4 & 1\end{bmatrix}\begin{bmatrix}1 \\ 0 \\ - 1\end{bmatrix} = \begin{bmatrix}x\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}- 3 + 0 - 1\end{bmatrix} = \begin{bmatrix}x\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}- 4\end{bmatrix} = \begin{bmatrix}x\end{bmatrix}\]
The corresponding elements of two equal matrices are equal .
\[ \therefore x = - 4 \]
\[ \therefore A = \left[ - 4 \right]\]
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