Question
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
$${\text{Tr}}\left( A \right) = $$ sum of diagonal elements of $$A$$ and
$$\left| A \right| = $$ determinant of matrix $$A.$$
Statement - 1 : $${\text{Tr}}\left( A \right) = 0$$
Statement - 2 : $$\left| A \right| = 1.$$
A.
Statement - 1 is true, Statement - 2 is true ; Statement - 2 is not a correct explanation for Statement - 1.
B.
Statement - 1 is true, Statement - 2 is false.
C.
Statement - 1 is false, Statement - 2 is true .
D.
Statement - 1 is true, Statement - 2 is true ; Statement - 2 is a correct explanation for Statement - 1.
Answer :
Statement - 1 is true, Statement - 2 is false.
Solution :
\[{\rm{Let }}\,A = \left( \begin{array}{l}
a\,\,\,\,\,\,\,b\\
c\,\,\,\,\,\,\,d
\end{array} \right){\rm{where }}\,\,a,b,c,d \ne 0\]
\[{A^2} = \left( \begin{array}{l}
a\,\,\,\,\,\,b\\
c\,\,\,\,\,\,d
\end{array} \right)\left( \begin{array}{l}
a\,\,\,\,\,\,b\\
c\,\,\,\,\,\,d
\end{array} \right)\]
\[ \Rightarrow \,\,{A^2} = \left( \begin{array}{l}
{a^2} + bc\,\,\,\,\,\,\,\,\,\,ab + bd\\
ac + cd\,\,\,\,\,\,\,\,\,\,bc + {d^2}
\end{array} \right)\]
$$\eqalign{
& \Rightarrow \,\,{a^2} + bc = 1,bc + {d^2} = 1 \cr
& ab + bd = ac + cd = 0 \cr
& c \ne 0\,\,{\text{and }}b \ne 0 \cr
& \Rightarrow \,\,a + d = 0 \cr
& \Rightarrow \,\,{\text{Tr}}\left( A \right) = 0 \cr
& \left| A \right| = ad - bc = - {a^2} - bc = - 1 \cr} $$