Question

If $A=\left|\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right|$ and $I$ is the identity matrix of order $3$, then show that$A^3=pI+qA+r{A}^2$.

Answer 1

$A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right]$

Now,

$\Rightarrow$ $A^2=AA$

$\Rightarrow$ $A^2=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right]\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right]$

$\Rightarrow$ $A^2=\left[\begin{array}{ccc}0+0+0& 0+0+0& 0+1+0\\ 0+0+p& 0+0+q& 0+0+r\\ 0+0+rp& p+0+rq& 0+q+r^2\end{array}\right]$

$\Rightarrow$ $A^2=\left[\begin{array}{ccc}0& 0& 1\\ p& q& r\\ rp& p+rq& q+r^2\end{array}\right]$

$\Rightarrow$ $A^3=A^{2}A$

$\Rightarrow$ $A^3=\left[\begin{array}{ccc}0& 0& 1\\ p& q& r\\ rp& p+rq& q+r^2\end{array}\right]\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right]$

$\Rightarrow$ $A^3=\left[\begin{array}{ccc}0+0+p& 0+0+q& 0+0+r\\ 0+0+rp& p+0+rq& 0+q+r^2\\ 0+0+pq+r^{2}p& rp+0+q^2+r^{2}q& 0+p+rq+rq+r^3\end{array}\right]$

$\Rightarrow$ $A^3=\left[\begin{array}{ccc}p& q& r\\ rp& p+rq& q+r^2\\ pq+r^{2}p& rp+q^2+r^{2}q& p+2rq+r^3\end{array}\right]$ ---- ( 1 )

$\Rightarrow$ $pI+qA+r{A}^2=p\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+q\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right]+r\left[\begin{array}{ccc}0& 0& 1\\ p& q& r\\ rp& p+rq& q+r^2\end{array}\right]$

$\Rightarrow$ $pI+qA+r{A}^2=\left[\begin{array}{ccc}p& 0& 0\\ 0& p& 0\\ 0& 0& p\end{array}\right]+\left[\begin{array}{ccc}0& q& 0\\ 0& 0& q\\ pq& q^2& qr\end{array}\right]+\left[\begin{array}{ccc}0& 0& r\\ rp& rq& r^2\\ r^{2}p& rp+r^{2}q& rq+r^3\end{array}\right]$

$\Rightarrow$ $pI+qA+r{A}^2=\left[\begin{array}{ccc}p& q& r\\ rp& p+rq& q+r^2\\ pq+r^{2}p& q^2+r^{2}q+rp& p+2qr+r^3\end{array}\right]$ --- ( 2 )

From ( 1 ) and ( 2 )

$\Rightarrow$ $A^3=pI+qA+r{A}^2$