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

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

Answers (1)

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

Posted by

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