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Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $tr(A),$ the sum of diagonal entries of $a$. Assume that ${A^2} = I$ .
Statement $-1 :$ If $A \ne I,A \ne - I$ then $\det \left( A \right) = - 1$
Statement $-2 :$ If $A \ne I,A \ne - I$ then ${\rm{tr}}\left( A \right) \ne 0$
Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is not acorrect explanation for Statement $-1$
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 false
Solution
$A = \left[ {\begin{array}{*{20}{c}}
a&b\\
c&d
\end{array}} \right]$
${A^2} = \left[ {\begin{array}{*{20}{c}}
a&b\\
c&d
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
a&b\\
c&d
\end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}}
{{a^2} + bc}&{ab + bd}\\
{ac + cd}&{bc + {d^2}}
\end{array}} \right] = I$
${a^2} + bc = bc + {d^2} = 1$
$ac + cd = ab + bd = 0$
$ac + cd = ab + bd = 0$
$b\left( {a + d} \right) = 0$
$c = 0\;\:{\rm{or}}\;\:a = – d$ not possible for $c$
$b = 0\;\:{\rm{or}}\;\:a = – d$ not possible for $b$
$\left| {\begin{array}{*{20}{c}}
a&b\\
c&d
\end{array}} \right| = ad – bc = – {d^2} – bc$
$ = – \left( {{d^2} + bc} \right) = – 1$
$tr\left( A \right) = a + d = a – a = 0$
