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, show that A³ = pI + qA +rA².

Answer 1

$$\begin{gathered} \mathrm{Given}:A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right] \\ \mathrm{Now}, \\ {A}^2=AA \\ \Rightarrow A^2=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right]\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{ccc}0+0+0& 0+0+0& 0+1+0\\ 0+0+p& 0+0+q& 0+0+r\\ 0+0+rp& p+0+rq& 0+q+r^2\end{array}\right] \\ {A}^2=\left[\begin{array}{ccc}0& 0& 1\\ p& q& r\\ rp& p+rq& q+r^2\end{array}\right] \\ {A}^3=A^{2}A \\ \Rightarrow A^3=\left[\begin{array}{ccc}0& 0& 1\\ p& q& r\\ rp& p+rq& q+r^2\end{array}\right]\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right] \\ \Rightarrow A^3=\left[\begin{array}{ccc}0+0+p& 0+0+q& 0+0+r\\ 0+0+rp& p+0+rq& 0+q+r^2\\ 0+0+pq+r^{2}p& rp+0+q^2+r^{2}q& 0+p+rq+rq+r^3\end{array}\right] \\ \Rightarrow A^3=\left[\begin{array}{ccc}p& q& r\\ rp& p+rq& q+r^2\\ pq+r^{2}p& rp+q^2+r^{2}q& p+2rq+r^3\end{array}\right]...\left(1\right) \\ pI+qA+r{A}^2 \\ \Rightarrow pI+qA+r{A}^2=p\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+q\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right]+r\left[\begin{array}{ccc}0& 0& 1\\ p& q& r\\ rp& p+rq& q+r^2\end{array}\right] \\ \Rightarrow pI+qA+r{A}^2=\left[\begin{array}{ccc}p& 0& 0\\ 0& p& 0\\ 0& 0& p\end{array}\right]+\left[\begin{array}{ccc}0& q& 0\\ 0& 0& q\\ pq& q^2& qr\end{array}\right]+\left[\begin{array}{ccc}0& 0& r\\ rp& rq& r^2\\ r^{2}p& rp+r^{2}q& rq+r^3\end{array}\right] \\ \Rightarrow pI+qA+r{A}^2=\left[\begin{array}{ccc}p+0+0& 0+q+0& 0+0+r\\ 0+0+rp& p+0+rq& 0+q+r^2\\ 0+pq+r^{2}p& 0+q^2+rp+r^{2}q& p+qr+qr+r^3\end{array}\right] \\ \Rightarrow pI+qA+r{A}^2=\left[\begin{array}{ccc}p& q& r\\ rp& p+rq& q+r^2\\ pq+r^{2}p& q^2+r^{2}q+rp& p+2qr+r^3\end{array}\right]...\left(2\right) \\ {A}^3=pI+qA+r{A}^2\left[\mathrm{From}\mathrm{eqs}.\left(1\right)\mathrm{and}\left(2\right)\right] \end{gathered}$$