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Let $ A$ be a $2$$ \times $$2$ matrix with non-zero entries and let ${A^2} = I$ where $I$ is $2\times 2$ identity matrix. Define $tr(A) =$ sum of diagonal elements of $A$ and $|A|=$ determinant of matrix $A$
Statement $-1 :$ ${\rm{tr}}\left( A \right) = 0$
Statement $-2 :$ $\det \left( A \right) = 1$
Statement $-1$ is false, Statement $-2$ is true;
Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is a correct explanation for Statement $-1$
Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is not acorrect explanation for Statement $-1$
Statement $-1$ is true, Statement $-2$ is false
Solution
Let $A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]$
It is given that,
$A^{2}=I$
${\therefore \left[ {\begin{array}{*{20}{c}}
a&b\\
c&d
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
a&b\\
c&d
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
1&0\\
0&1
\end{array}} \right]}$
${ \Rightarrow \left[ {\begin{array}{*{20}{c}}
{{a^2} + bc}&{ab + bd}\\
{ac + cd}&{bc + {d^2}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
1&0\\
0&1
\end{array}} \right]}$
${ \Rightarrow {a^2} + bc = 1 \to 1}$
${ \Rightarrow ab + bd = 0 \Rightarrow b(a + d) = 0 \Rightarrow a = – d \to (2) \ldots [Asb \ne 0]}$
So, we can write
$A=\left[\begin{array}{cc}{a} & {b} \\ {c} & {-a}\end{array}\right]$
$\therefore \operatorname{Tr}(A)=a+(-a)=0$
$|A|=-a^{2}-b c=-\left(a^{2}+b c\right)=-1$
So, first statement is true but second statement is false.
