#### Provide solution for rd sharma math class class 12 chapter Algebra of matrices exercise 4.3 question 3 sub question (ix)

$A^{2}=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] and \ A^{3}=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]$

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: $A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$

Consider,$A^2 = AA$

$A^{2}=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}1 \times 1+1 \times 0 & 1 \times 1+1 \times 1 \\ 0 \times 1+1 \times 0 & 0 \times 1+1 \times 1\end{array}\right] \\\\ \\A^{2}=\left[\begin{array}{ll}1+0 & 1+1 \\ 0+0 & 0+1\end{array}\right]=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$

Again consider

\begin{aligned} A^{3} &=A^{2} A \\\\ A^{3} &=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{ll} 1 \times 1+2 \times 0 & 1 \times 1+2 \times 1 \\ 0 \times 1+1 \times 0 & 0 \times 1+1 \times 1 \end{array}\right] \\\\ A^{3} &=\left[\begin{array}{ll} 1+0 & 1+2 \\ 0+0 & 0+1 \end{array}\right] \\ \\A^{3} &=\left[\begin{array}{ll} 1 & 3 \\ 0 & 1 \end{array}\right] \end{aligned}

Hence, proved.