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Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices1 Mark
MCQ
If $\text{A}=\begin{bmatrix}\text{i}&0\\0&\text{i}\end{bmatrix},\text{n}\in\text{N},$ then $A^{4n}$ equals:
  • A
    $\begin{bmatrix}0&\text{i}\\\text{i}&0\end{bmatrix}$
  • B
    $\begin{bmatrix}0&0\\0&0\end{bmatrix}$
  • ✓
    $\begin{bmatrix}1&0\\0&1\end{bmatrix}$
  • D
    $\begin{bmatrix}0&\text{i}\\\text{i}&0\end{bmatrix}$
✓

Answer

Correct option: C.
$\begin{bmatrix}1&0\\0&1\end{bmatrix}$
Given $\text{A}=\begin{bmatrix}\text{i}&0\\0&\text{i}\end{bmatrix}=\text{i}\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$\text{A}^4=\text{i}^4\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$=\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$\text{A}^{4\text{n}}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$

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