Find the value of x for which the matrix product

Question:

Find the value of x for which the matrix product
$\left[\begin{array}{rrr}2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1\end{array}\right]\left[\begin{array}{rrr}-x & 14 x & 7 x \\ 0 & 1 & 0 \\ x & -4 x & -2 x\end{array}\right]$              equal an identity matrix.

Solution:

Here,

$\left[\begin{array}{ccc}2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1\end{array}\right]\left[\begin{array}{ccc}-x & 14 x & 7 x \\ 0 & 1 & 0 \\ x & -4 x & -2 x\end{array}\right]=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

$\Rightarrow\left[\begin{array}{ccc}-2 x+0+7 x & 28 x+0-28 x & 14 x+0-14 x \\ 0+0+0 & 0+1-0 & 0+0-0 \\ -x-0+x & 14 x-2-4 x & 7 x-0-2 x\end{array}\right]$ $=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

$\Rightarrow\left[\begin{array}{ccc}5 x & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 10 x-2 & 5 x\end{array}\right]=$ $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

The corresponding elements of two equal matrices are equal.

$\therefore 5 x=1$

$\Rightarrow x=\frac{1}{5}$

 

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