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 A be a 2 × 2 matrix with non-zero entries and let  A^{2}=I,  where I is 2 × 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

 

 

 

  • 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.

 

Answers (2)

As we learnt in 

Multiplication of matrices -

-

 

 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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