Let a be a `2xx2` matrix with non-zero entries and let `A^(2)=I`, where `I` is a `2xx2` 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`

Options

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

B

Explanation

Let `A = ((a,b),(c,d)), abcd ne 0`
`A^(2) = ((a,b),(c,d)) cdot ((a,b),(c,d))`
`rArr A ^(2) = ((a^(2)+bc,ab+bd),(ac+cd,bc+d^(2)))`
`rArr a^(2) + bc = 1, bc + d^(2) = 1`
` ab + bd = ac + cd = 0 `
`c ne 0 and bne 0 `
and `a + d= 0`
Trace A = a +d = 0`
`abs(A) = ad - bc = - a^(2) - bc = 1`