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#### Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 39 maths textbook solution

Answer: $x=1 / 5$

Hint: $I_{3}$ is an identity matrix of size

$3, I_{3}=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

Given:$\left[\begin{array}{ccc}2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1\end{array}\right]\left[\begin{array}{ccc}-x & 14 x & 7 x \\ 0 & 1 & 0 \\ x & -4 x & -2 x\end{array}\right]$ equal to an identity matrix

So, according to given criteria

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

Now, we will multiply the two matrices in LHS we get

$=\left[\begin{array}{ccc}-2 x+0+7 x & 2(14 x)+0+7(-4 x) & 2(7 x)+0+7(-2 x) \\ 0+0+0 & 0+1(1)+0 & 0+0+0 \\ 1(-x)+0+1(x) & 1(14 x)+(-2)(1)+1(-4 x) & 1(7 x)+0+1(-2 x)\end{array}\right] \\\\\\ =\left[\begin{array}{ccc}5 x & 28 x-28 x & 14 x-14 x \\ 0 & 0+1+0 & 0\end{array}\right]\\\\$

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

LHS=RHS (given)

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

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal.

So, we get

5x=1

x=1/5

So, the value of x is 1/5