Question

If $A$ is matrix that $A^2=A+I$, where $I$ is unit matrix then $A^5$ is equal to

• A. $5A+I$
• B. $5A+2I$
• C. $5A+3I$
• D. $5A+4I$

Answer 1

The correct option is C

$5A+3I$

Explanation for the correct option:

Find the value of $A^5$:

Given, $A^2=A+I$

$\begin{array}{rcl}{A}^5& =& A^{4}A\\ & =& A^2{A}^{2}A\end{array}$

$\begin{array}{rcl}& =& \left[A+I\right]\left[A+I\right]A\end{array}$

$\begin{array}{rcl}& =& \left[A^2+\hspace{0.17em}{I}^2+2AI\right]A\end{array}$

$\begin{array}{rcl}& =& \left[A+I+2A+I\right]A\end{array}$ $\left[\mathbf{\because }\mathbf{2}\mathit{A}\mathit{I}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}\mathit{A}\mathbf{,}\mathbf{}{\mathit{I}}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\mathit{I}\right]$

$\begin{array}{rcl}& =& \left[3A+2I\right]A\end{array}$

$\begin{array}{rcl}& =& 3A^2+2AI\end{array}$

$=3\left(A+I\right)+2A$ $\left[\because A^2=A+I\right]$

$=3A+3I+\hspace{0.17em}2A$

$\mathbf{\therefore }{\mathit{A}}^{\mathbf{5}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{5}\mathit{A}\mathbf{}\mathbf{+}\mathbf{}\mathbf{3}\mathit{I}$

Hence, Option ‘C’ is Correct.