# Directions : Question are Assertion- Reason type questions. Each of these questions contains two statements. Statement-1 (Assertion) and Statement-2 (Reason). Each of these questions also has four alternative choices,only one of which is the correct answer. You have to select the correct choice:Question : Let $\dpi{100} A$ be a 2 × 2 matrix with non-zero entries and let  $\dpi{100} A^{2}=I,$  where $\dpi{100} I$ is 2 × 2 identity matrix.   Define        $\dpi{100} Tr(A)=$   sum of diagonal elements of  $\dpi{100} A$ and $\dpi{100} \left | A \right |$ = determinant of matrix $\dpi{100} A$.Statement-1 : $\dpi{100} Tr(A)=$ 0Statement-2 : $\dpi{100} \left | A \right |$ = 1 Option 1) Statement -1 is true, statement -2 is true ; statement -2 is a correct explanation of statement-1. Option 2) Statement -1 is true, statement -2 is true ; statement -2 is not a correct explanation of statement-1. Option 3) Statement -1 is true, statement -2 is false. Option 4) Statement -1 is false, statement -2 is true.

N neha
P Prateek Shrivastava

As we learnt in

Multiplication of matrices -

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$A^{2}= I\, \, given\, \, so\, \, that$

$a^{2}+bc= 1,\, \, ab+bd= 0,\, \,ac+cd= 0,\, bc+d^{2}=1$

$\therefore b= 0, \, \, \, \, \, \, \, \ a+d=0$

$If \, \, b=0, \, \, \, d^{2}=1, d=\pm 1$

$If \, \, c=0 \, \, \, \ a^{2}=1, a=\pm 1$

$\therefore A=\begin{vmatrix}-1&0 \\0&1 \end{vmatrix}\, \, \, \, \, or\, \, \, \, \, A=\begin{vmatrix}1&0 \\0&-1 \end{vmatrix}$

$\therefore tr\left ( A \right )= 0 and |A|= -1$

Option 1)

Statement -1 is true, statement -2 is true ; statement -2 is a correct explanation of statement-1.

Incorrect Option

Option 2)

Statement -1 is true, statement -2 is true ; statement -2 is not a correct explanation of statement-1.

Incorrect Option

Option 3)

Statement -1 is true, statement -2 is false.

Correct Option

Option 4)

Statement -1 is false, statement -2 is true.

Incorrect Option

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