Question

If $A$ is a $2\times 2$ matrix with non - zero entries let $A^2=I$, where $I$ is $2\times 2$ identity matrix.

Define $tr\left(A\right)=$ Sum of diagonal element of $A$ and $\left|A\right|=$ Determinant of matrix $A$.

Statement I $tr\left(A\right)=0$.

Statement II $\left|A\right|=i$

• A. Statement I is correct, Statement II is correct; Statement II is the correct explanation for Statement I
• B. Statement I is correct, statement II is correct, Statement II is not the correct explanation for Statement I
• C. Statement I is correct, Statement II is incorrect
• D. Statement I is incorrect, Statement II is correct

Answer 1

The correct option is C

Statement I is correct, Statement II is incorrect

Explanation for the correct option:

Given, $A$ is a $2\times 2$ matrix.

$A^2=I$ where $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$A=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$

$A^2=A\times A$

$A^2=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\times \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\begin{array}{rcl}{A}^2& =& \left[\begin{array}{cc}{\mathrm{a}}^2+\mathrm{bc}& \mathrm{ab}+\mathrm{bd}\\ \mathrm{ac}+\mathrm{cd}& \mathrm{bc}+{\mathrm{d}}^2\end{array}\right]\\ & =& \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$

$$\begin{gathered} \therefore a^2+bc=1;ab+bd=0 \\ bc+a^2=1;ac+cd=0 \end{gathered}$$

Here, $c\ne 0$ and $b\ne 0$

$\therefore a+d=0$

$\begin{array}{rcl}tr\left(A\right)& =& a+d\\ & =& 0\end{array}$

$\begin{array}{rcl}\left|A\right|& =& ad-bc\\ & =& -a^2-bc\\ & =& -\left(a^2+bc\right)\end{array}$

$\therefore \left|A\right|\mathbf{=}\mathbf{-}\mathbf{1}$

Hence, option(C) is correct.