JEE MAIN - Mathematics (2010 - No. 12)
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 $$\left| A \right| = $$ determinant of matrix $$A$$.
Statement- 1: $$Tr$$$$(A)=0$$.
Statement- 2: $$\left| A \right| = 1$$ .
where $$I$$ is $$2 \times 2$$ identity matrix. Define
$$Tr$$$$(A)=$$ sum of diagonal elements of $$A$$ and $$\left| A \right| = $$ determinant of matrix $$A$$.
Statement- 1: $$Tr$$$$(A)=0$$.
Statement- 2: $$\left| A \right| = 1$$ .
statement - 1 is true, statement - 2 is true; statement - 2 is not a correct explanation for statement - 1.
statement - 1 is true, statement - 2 is false.
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.
Explanation
Let $$A = \left( {\matrix{
a & b \cr
c & d \cr
} } \right)$$ where $$a,b,c,d$$ $$ \ne 0$$
$${A^2} = \left( {\matrix{ a & b \cr c & d \cr } } \right)\left( {\matrix{ a & b \cr c & d \cr } } \right)$$
$$ \Rightarrow {A^2} = \left( {\matrix{ {{a^2} + bc} & {ab + bd} \cr {ac + cd} & {bc + {d^2}} \cr } } \right)$$
$$ \Rightarrow {a^2} + bc = 1,\,bc + {d^2} = 1$$
$$ab + bd = ac + cd = 0$$
$$c \ne 0\,\,\,\,\,b \ne 0$$
$$ \Rightarrow a + d = 0 \Rightarrow Tr\left( A \right) = 0$$
$$\left| A \right| = ad - bc = - {a^2} - bc = - 1$$
$${A^2} = \left( {\matrix{ a & b \cr c & d \cr } } \right)\left( {\matrix{ a & b \cr c & d \cr } } \right)$$
$$ \Rightarrow {A^2} = \left( {\matrix{ {{a^2} + bc} & {ab + bd} \cr {ac + cd} & {bc + {d^2}} \cr } } \right)$$
$$ \Rightarrow {a^2} + bc = 1,\,bc + {d^2} = 1$$
$$ab + bd = ac + cd = 0$$
$$c \ne 0\,\,\,\,\,b \ne 0$$
$$ \Rightarrow a + d = 0 \Rightarrow Tr\left( A \right) = 0$$
$$\left| A \right| = ad - bc = - {a^2} - bc = - 1$$
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