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