#### Provide solution for rd sharma math class 12 chapter Algebra of matrices exercise 4.3 question 41

$A^{2}-4 A+3 I_{3}=\left[\begin{array}{ccc}6 & -14 & 10 \\ -21 & 36 & -25 \\ -3 & 5 & -5\end{array}\right]$

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:

$A=\left[\begin{array}{ccc}1 & 2 & 0 \\ 3 & -4 & 5 \\ 0 & -1 & 3\end{array}\right]$

$I_3$ is identity matrix of size 3.

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

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

$\begin {array}{ll}A^{2}=A A=\left[\begin{array}{ccc}1 & 2 & 0 \\ 3 & -4 & 5 \\ 0 & -1 & 3\end{array}\right]\left[\begin{array}{ccc}1 & 2 & 0 \\ 3 & -4 & 5 \\ 0 & -1 & 3\end{array}\right] \\\\\ \\A^{2}=\left[\begin{array}{ccc}1+6+0 & 2-8+0 & 0+10+0 \\ 3-12+0 & 6+16-5 & 0-20+15 \\ 0-3+0 & 0+4-3 & 0-5+9\end{array}\right] \\\\\\A^{2}=\left[\begin{array}{ccc}7 & -6 & 10 \\ -9 & 17 & -5 \\ -3 & 1 & 4\end{array}\right] \ \ \ \ ...(i) \end{}$

Now, we will find the matrix for 4A, we get

$\begin {array}{ll} 4 A=4\left[\begin{array}{ccc}1 & 2 & 0 \\ 3 & -4 & 5 \\ 0 & -1 & 3\end{array}\right] \\\\\ 4 A=\left[\begin{array}{ccc}4 & 8 & 0 \\ 12 & -16 & 20 \\ 0 & -4 & 12\end{array}\right] \ \ \ ...(ii) \end{}$

Substituting corresponding values from equation i & ii in the given equation, we get$A^{2}-4 A+3 I_{3} \\\\ =\left[\begin{array}{ccc}7 & -6 & 10 \\ -9 & 17 & -5 \\ -3 & 1 & 4\end{array}\right]-\left[\begin{array}{ccc}4 & 8 & 0 \\ 12 & -16 & 20 \\ 0 & -4 & 12\end{array}\right]+\left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{array}\right] \\\\\\ =\left[\begin{array}{ccc}7-4+3 & -6-8+0 & 10-0+0 \\ -9-12+0 & 17+16+3 & -5-20+0 \\ -3-0+0 & 1+4+0 & 4-12+3\end{array}\right] \\\\\\ =\left[\begin{array}{ccc}6 & -14 & 10 \\ -21 & 36 & -25 \\ -3 & 5 & -5\end{array}\right]$

Hence,

$A^{2}-4 A+3 I_{3}=\left[\begin{array}{ccc}6 & -14 & 10 \\ -21 & 36 & -25 \\ -3 & 5 & -5\end{array}\right]$