If A= 0&1&0 0&0&1 &q&r , and I is the identity matrix of order 3, show that A^3 = pI + qA + rA^2.

Given, A= 0&1&0 0&0&1 &q&r , A^2=A = 0&1&0 0&0&1 &q&r 0&1&0 0&0&1 &q&r 0+0+0&0+0+0&0+1+0 0+0+p&0+0+q&0+0+r 0+0+pr&p+0+qr&0+q+r^2 A^3=A^2 = 0&0&1 &q&r &p+qr&q+r^2 0&1&0 0&0&1 &q&r = 0+0+p&0 +0+q&0+0+r 0+0+pr&p+0+qr&0+q+r^2 0+0+pq+pr^2&pr+0+q^2+qr^2&0+p+qr+qr+r^2 A^3= p&q&r &p+qr&q+r^2 +pr^2&pr+q^2+qr^2&p+2qr+r^2 pI+qA+rA^2 =p 1&0&0 0&1&0 0&0&1 +q 0&1&0 0&0&1 &q&r +r 0&0&1 &q&r &p+qr&q+r^2 = p+0+0&0+q+0&0+0+r 0+0+pr&p+0+qr&0+q+r^2 0+pq+pr^2&0+q^2+pr+qr^2&p+qr+qr+r^2 pI+qA+rA^2 = p&q&r &p+pr&q+r^2 +pr^2&pr+q^2+qr^2&p+2qr+r^2

If A= 0&1&0 0&0&1 &q&r , and I is the identity matrix of order 3,…

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Question

If $\text{A}=\begin{bmatrix}0&1&0\\0&0&1\\\text{p}&\text{q}&\text{r}\end{bmatrix},$ and I is the identity matrix of order $3,$ show that $A^3 = pI + qA + rA^2.$

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Answer

Given,
$\text{A}=\begin{bmatrix}0&1&0\\0&0&1\\\text{p}&\text{q}&\text{r}\end{bmatrix},$
$\text{A}^2=\text{A}\times\text{A}$
$=\begin{bmatrix}0&1&0\\0&0&1\\\text{p}&\text{q}&\text{r}\end{bmatrix}\begin{bmatrix}0&1&0\\0&0&1\\\text{p}&\text{q}&\text{r}\end{bmatrix}$
$\begin{bmatrix}0+0+0&0+0+0&0+1+0\\0+0+\text{p}&0+0+\text{q}&0+0+\text{r}\\0+0+\text{pr}&\text{p}+0+\text{qr}&0+\text{q}+\text{r}^2\end{bmatrix}$
$\text{A}^3=\text{A}^2\times\text{A}$
$=\begin{bmatrix}0&0&1\\\text{p}&\text{q}&\text{r}\\\text{pr}&\text{p}+\text{qr}&\text{q}+\text{r}^2\end{bmatrix}\begin{bmatrix}0&1&0\\0&0&1\\\text{p}&\text{q}&\text{r}\end{bmatrix}$
$=\begin{bmatrix}0+0+\text{p}&0 +0+\text{q}&0+0+\text{r}\\0+0+\text{pr}&\text{p}+0+\text{qr}&0+\text{q}+\text{r}^2\\0+0+\text{pq}+\text{pr}^2&\text{pr}+0+\text{q}^2+\text{qr}^2&0+\text{p}+\text{qr}+\text{qr}+\text{r}^2\end{bmatrix}$
$\text{A}^3= \begin{bmatrix}\text{p}&\text{q}&\text{r}\\\text{pr}&\text{p}+\text{qr}&\text{q}+\text{r}^2\\\text{pq}+\text{pr}^2&\text{pr}+\text{q}^2+\text{qr}^2&\text{p}+2\text{qr}+\text{r}^2\end{bmatrix}$
$\text{pI}+\text{qA}+\text{rA}^2$
$=\text{p} \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}+\text{q}\begin{bmatrix}0&1&0\\0&0&1\\\text{p}&\text{q}&\text{r}\end{bmatrix}+\text{r}\begin{bmatrix}0&0&1\\\text{p}&\text{q}&\text{r}\\\text{pr}&\text{p}+\text{qr}&\text{q}+\text{r}^2\end{bmatrix}$
$= \begin{bmatrix}\text{p}+0+0&0+\text{q}+0&0+0+\text{r}\\0+0+\text{pr}&\text{p}+0+\text{qr}&0+\text{q}+\text{r}^2\\0+\text{pq}+\text{pr}^2&0+\text{q}^2+\text{pr}+\text{qr}^2&\text{p}+\text{qr}+\text{qr}+\text{r}^2\end{bmatrix}$
$\text{pI}+\text{qA}+\text{rA}^2$
$= \begin{bmatrix}\text{p}&\text{q}&\text{r}\\\text{pr}&\text{p}+\text{pr}&\text{q}+\text{r}^2\\\text{pq}+\text{pr}^2&\text{pr}+\text{q}^2+\text{qr}^2&\text{p}+2\text{qr}+\text{r}^2\end{bmatrix}$

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