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#### Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 35 maths textbook solution

Hint: Matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.

Given: $A=\left[\begin{array}{ll} 3 & -2 \\ 4 & -2 \end{array}\right]$ and $A^{2}=k A-2 I_{2}$

$I_ 2$ is an identity matrix of size 2. So, $2 I_{2}=2\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right]$

Also given,

$A^{2}=k A-2 I_{2}$

Now, we will find the matrix for $A^2$, we get

$A^{2}=A A=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right] \\\\ A^{2}=\left[\begin{array}{cc}9-8 & -6+4 \\ 12-8 & -8+4\end{array}\right] \\\\\ A^{2}=\left[\begin{array}{ll}1 & -2 \\ 4 & -4\end{array}\right]----i$

Now, we will find the value for kA, we get

$k A=k\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right] \\\\ k A=\left[\begin{array}{ll}k \times 3 & k \times(-2) \\ k \times 4 & k \times(-2)\end{array}\right]$

So, substituting corresponding values from equation i & ii in
$A^{2}=k A-2 I_{2}$

we get$\left[\begin{array}{ll}1 & -2 \\ 4 & -4\end{array}\right]=\left[\begin{array}{ll}3 k & -2 k \\ 4 k & -2 k\end{array}\right]-\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right] \\\\\ \\ \left[\begin{array}{ll}1 & -2 \\ 4 & -4\end{array}\right]=\left[\begin{array}{ll}3 k-2 & -2 k-0 \\ 4 k-0 & -2 k-2\end{array}\right]$

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

$\begin{array}{l} 3 k=1+2 \\\\ 3 k=3 \\\\ k=\frac{3}{3}=1 \end{array}$

Therefore, the value of k is 1