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Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices1 Mark
MCQ
If $\text{A}=\begin{bmatrix}1 \text{a}0 1\end{bmatrix},$ then $A^n ($where $n \in N)$ equals:
  • ✓
    $\begin{bmatrix}1&\text{na}\\0&1\end{bmatrix}$
  • B
    $\begin{bmatrix}1&\text{n}^2\text{a}\\0&1\end{bmatrix}$
  • C
    $\begin{bmatrix}1&\text{n}\text{a}\\0&0\end{bmatrix}$
  • D
    $\begin{bmatrix}\text{n}&\text{n}\text{a}\\0&\text{n}\end{bmatrix}$
✓

Answer

Correct option: A.
$\begin{bmatrix}1&\text{na}\\0&1\end{bmatrix}$
Given $\text{A}=\begin{bmatrix}1&\text{a}\\0&1\end{bmatrix}$
$\text{A}^2=\begin{bmatrix}1&\text{a}\\0&1\end{bmatrix}\begin{bmatrix}1&\text{a}\\0&1\end{bmatrix}$
$=\begin{bmatrix}1&2\text{a}\\0&1\end{bmatrix}$
$\text{A}^3=\text{A}^2\text{A}$
$=\begin{bmatrix}1&2\text{a}\\0&1\end{bmatrix}\begin{bmatrix}1&\text{a}\\0&1\end{bmatrix}$
$=\begin{bmatrix}1&3\text{a}\\0&1\end{bmatrix}$
On genaralising we get
$\text{A}^\text{n}=\begin{bmatrix}1&\text{na}\\0&1\end{bmatrix}$

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