Let $A$ be a square matrix such that $AA^T = I$. Then $\frac{1}{2} A[(A+A^T)^2 + (A-A^T)^2]$ is equal to

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible if

If $A$ is a skew-symmetric matrix of order $3$ and $X$ is another matrix of the same order,then $|XA + AX^T|$ is (where $|P|$ denotes the determinant of matrix $P$).

If $x, y$ are any two non-zero real numbers, $a_{i j} = xi + yj$, $A = \{a_{i j}\}_{n \times n}$ and $P, Q$ are two $n \times n$ matrices such that $A = xP + yQ$, then

Let $A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$,where $x, y$ and $z$ are real numbers such that $x + y + z > 0$ and $xyz = 2$. If $A^2 = I_3$,then the value of $x^3 + y^3 + z^3$ is ............

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $tr(A)$ the sum of diagonal entries of $A$. Assume that $A^2 = I$.
Statement-$1$: If $A \neq I$ and $A \neq -I$,then $\det(A) = -1$.
Statement-$2$: If $A \neq I$ and $A \neq -I$,then $tr(A) \neq 0$.