#### Please Solve R.D. Sharma class 12 Chapter 4 Algbra of Matrices Exercise Multiple Choice Questions Question  2 Maths textbook Solution.

Answer: The correct option is $\text { (C), } A^{4}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$

Given: $A=\left[\begin{array}{ll} i & 0 \\ 0 & i \end{array}\right]$

Solution:

Now,  $A^{2}=A \times A$

\begin{aligned} &=\left[\begin{array}{ll} i & 0 \\ 0 & i \end{array}\right] \times\left[\begin{array}{cc} i & 0 \\ 0 & i \end{array}\right] \\ &\quad=\left[\begin{array}{cc} i^{2} & 0 \\ 0 & i^{2} \end{array}\right] \\ &=\left[\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right] \\ &A^{3}=A^{2} \times A \\ &=\left[\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right] \times\left[\begin{array}{cc} i & 0 \\ 0 & i \end{array}\right] \\ &=\left[\begin{array}{cc} -i & 0 \\ 0 & -i \end{array}\right] \end{aligned}

\begin{aligned} A^{4} &=A^{3} \times A \\ &=\left[\begin{array}{cc} -i & 0 \\ 0 & -i \end{array}\right] \times\left[\begin{array}{cc} i & 0 \\ 0 & i \end{array}\right] \\ &=\left[\begin{array}{cc} -i^{2} & 0 \\ 0 & -i^{2} \end{array}\right] \\ &=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ \text { So, } & A^{4}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}