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 Tr(A) $= 0$ Statement-2: $| A | = 1$

Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1

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Statement-1 is true, Statement-2 is false

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Statement-1 is false, Statement-2 is true

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Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

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Solution

The correct option is A Statement-1 is true, Statement-2 is false
Let $A = (\begin{matrix} a & b c & d \end{matrix})$ , abcd $\neq 0$
$A^{2} = (\begin{matrix} a & b c & d \end{matrix}) \cdot (\begin{matrix} a & b c & d \end{matrix})$
$A^{2} = (\begin{matrix} a^{2} + b c & a b + b d a c + c d & b c + d^{2} \end{matrix})$
$= a^{2} +$ bc $= 1$ , bc $+ d^{2} = 1$
$a b + b d = a c + c d = 0$
$c \neq 0$ and $b \neq 0 = a + d = 0$
Trace $A = a + d = 0 \Rightarrow a = - d$
$| A | = a d -$ bc $= - a^{2} -$ bc $= - 1$
Hence, option 'B' is correct.

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Let A be a 2×2 matrix with non-zero entries and let A2=I, where I is 2×2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A|= determinant of matrix A.Statement-1 Tr(A) =0 Statement-2: |A|=1

The correct option is A Statement-1 is true, Statement-2 is falseLet A=(abcd), abcd ≠0A2=(abcd)⋅(abcd)A2=(a2+bcab+bdac+cdbc+d2)=a2+ bc =1, bc +d2=1ab+bd=ac+cd=0c≠0 and b≠0=a+d=0Trace A=a+d=0⇒a=−d|A|=ad− bc =−a2− bc =−1 Hence, option 'B' is correct.

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 Tr(A) $= 0$ Statement-2: $| A | = 1$