If $A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r\end{array}\right]$, and $I$ is the identity matrix of order 3 , show that $A^{3}=p \mid+q A+r A^{2}$.
Given : $A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r\end{array}\right]$
Now,
$A^{2}=A A$
$\Rightarrow A^{2}=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r\end{array}\right]\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r\end{array}\right]$
$\Rightarrow A^{2}=\left[\begin{array}{lll}0+0+0 & 0+0+0 & 0+1+0 \\ 0+0+p & 0+0+q & 0+0+r \\ 0+0+r p & p+0+r q & 0+q+r^{2}\end{array}\right]$
$\begin{aligned} A^{2} &=\left[\begin{array}{ccc}0 & 0 & 1 \\ p & q & r \\ r p & p+r q & q+r^{2}\end{array}\right] \\ A^{3} &=A^{2} A \end{aligned}$
$\Rightarrow A^{3}=\left[\begin{array}{ccc}0 & 0 & 1 \\ p & q & r \\ r p & p+r q & q+r^{2}\end{array}\right]\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r\end{array}\right]$
$\Rightarrow A^{3}=\left[\begin{array}{ccc}0+0+p & 0+0+q & 0+0+r \\ 0+0+r p & p+0+r q & 0+q+r^{2} \\ 0+0+p q+r^{2} p & r p+0+q^{2}+r^{2} q & 0+p+r q+r q+r^{3}\end{array}\right]$
$\Rightarrow A^{3}=\left[\begin{array}{ccc}p & q & r \\ r p & p+r q & q+r^{2} \\ p q+r^{2} p & r p+q^{2}+r^{2} q & p+2 r q+r^{3}\end{array}\right]$
$p I+q A+r A^{2}$
$\Rightarrow p I+q A+r A^{2}=p\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]+q\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r\end{array}\right]+r\left[\begin{array}{ccc}0 & 0 & 1 \\ p & q & r \\ r p & p+r q & q+r^{2}\end{array}\right]$
$\Rightarrow p I+q A+r A^{2}=\left[\begin{array}{lll}p & 0 & 0 \\ 0 & p & 0 \\ 0 & 0 & p\end{array}\right]+\left[\begin{array}{ccc}0 & q & 0 \\ 0 & 0 & q \\ p q & q^{2} & q r\end{array}\right]+\left[\begin{array}{ccc}0 & 0 & r \\ r p & r q & r^{2} \\ r^{2} p & r p+r^{2} q & r q+r^{3}\end{array}\right]$
$\Rightarrow p I+q A+r A^{2}=\left[\begin{array}{ccc}p+0+0 & 0+q+0 & 0+0+r \\ 0+0+r p & p+0+r q & 0+q+r^{2} \\ 0+p q+r^{2} p & 0+q^{2}+r p+r^{2} q & p+q r+q r+r^{3}\end{array}\right]$
$\Rightarrow p I+q A+r A^{2}=\left[\begin{array}{ccc}p & q & r \\ r p & p+r q & q+r^{2} \\ p q+r^{2} p & q^{2}+r^{2} q+r p & p+2 q r+r^{3}\end{array}\right]$ ....(2)
$A^{3}=p I+q A+r A^{2}$ [From eqs. (1) and (2)]