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Question
Find the value of x for which the matrix product`[[2 0 7],[0 1 0],[1 -2 1]]` `[[-x 14x 7x],[0 1 0],[x -4x -2x]]`equal an identity matrix.
Solution
\[Here, \]
\[ \begin{bmatrix}2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & - 2 & 1\end{bmatrix}\begin{bmatrix}- x & 14x & 7x \\ 0 & 1 & 0 \\ x & - 4x & - 2x\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}- 2x + 0 + 7x & 28x + 0 - 28x & 14x + 0 - 14x \\ 0 + 0 + 0 & 0 + 1 - 0 & 0 + 0 - 0 \\ - x - 0 + x & 14x - 2 - 4x & 7x - 0 - 2x\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}5x & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 10x - 2 & 5x\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\]
\[\]
The corresponding elements of two equal matrices are equal .
\[ \therefore 5x = 1 \]
\[ \Rightarrow x = \frac{1}{5} \]
\[\]
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