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Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices5 Marks
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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