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Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices5 Marks
Question
If $\text{A}=\begin{bmatrix}1&1\\0&1\end{bmatrix},$ show that $\text{A}^2=\begin{bmatrix}1&2\\0&1\end{bmatrix}$ and $\text{A}^3=\begin{bmatrix}1&3\\0&1\end{bmatrix}.$
✓

Answer

Given, $\text{A}=\begin{bmatrix}1&1\\0&1\end{bmatrix}$
Now,
$\text{A}^2=\text{AA}$
$\Rightarrow\text{A}^2=\begin{bmatrix}1&1\\0&1\end{bmatrix}\begin{bmatrix}1&1\\0&1 \end{bmatrix}$
$\Rightarrow\text{A}^2=\begin{bmatrix}1+0&1+1\\0+0&0+1\end{bmatrix}$
$\Rightarrow\text{A}^2=\begin{bmatrix}1&2\\0&1\end{bmatrix}$
$ \text{A}^3=\text{A}^2\text{A}$
$\Rightarrow\text{A}^3=\begin{bmatrix}1&2\\0&1 \end{bmatrix}\begin{bmatrix}1&1\\0&1\end{bmatrix}$
$\Rightarrow\text{A}^3=\begin{bmatrix}1+0&1+2\\0+0&0+1 \end{bmatrix}$
$\Rightarrow\text{A}^3=\begin{bmatrix}1&3\\0&1 \end{bmatrix}$
Hence proved.

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