Let A be a 2x2 matrix with non-zero entries and let A²=I, where I is 2x2 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

Let A =
\begin{pmatrix} a & b \ c & d \end{pmatrix}, abcd \neq 0
Since A² = I, we have
\begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} a & b \ c & d \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}
\begin{pmatrix} a² + bc & ab + bd \ ac + cd & bc + d² \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}
This gives us the equations:
a² + bc = 1
ab + bd = 0
ac + cd = 0
bc + d² = 1
From the second equation, b(a+d) = 0. Since b ≠ 0, a + d = 0, so Tr(A) = a + d = 0. Statement 1 is true.
Also, |A| = ad - bc. We have (a+d)² = a² + 2ad + d² = 0, so a² + d² = -2ad.
From the equations above, a² + bc = 1 and bc + d² = 1. Therefore,
1 - bc + 1 - bc = -2ad
2 - 2bc = -2ad
1 - bc = -ad
ad - bc = 1
Thus, |A| = 1. Statement 2 is true. However, statement 2 is not the correct explanation for statement 1.
Therefore, the correct option is: Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1