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Let A be a 2 xx 2 matrix with non-zero e...

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

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

Text Solution

Verified by Experts

The correct Answer is:
B
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`
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