1. Tardigrade
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  3. Mathematics
  4. Let A be a 2 × 2 matrix with non-zero entries and let A2 = I, where I is 2 × 2 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

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Q. Let $A$ be a $2 \times 2$ matrix with non-zero entries and let $A^2$ = $I$, where $I$ is $2 \times 2$ identity matrix. Define $Tr (A)$ = Sum of diagonal elements of $A$ and $|A|$ = determinant of matrix $A$.
Statement-1. $Tr (A) = 0$
Statement-2. $|A| = 1$

AIEEEAIEEE 2008Determinants

Solution:

Let A = $\begin{bmatrix}a&b\\ c&d\end{bmatrix}$ $\therefore $ adj A =$\begin{bmatrix}a&b\\ c&d\end{bmatrix}$
$\Rightarrow $ adj (adj A) = $\begin{bmatrix}a&b\\ c&d\end{bmatrix} = A$ $\therefore $ Statement-1 is true.
Also | adj A | = ad - bc = 1 | A |
$\therefore $ Statement-2 is true.
Since Statement-2 does not give Statement-1