Let $$P$$ and $$Q$$ be $$3 \times 3$$ matrices $$P \ne Q.$$ If $${P^3} = {Q^3}$$ and
$${P^2}Q = {Q^2}P$$ then determinant of $$\left( {{P^2} + {Q^2}} \right)$$ is equal to :

Let $$A$$ and $$B$$ be two symmetric matrices of order $$3$$.

Statement - 1 : $$A(BA)$$ and $$(AB)$$$$A$$ are symmetric matrices.

Statement - 2 : $$AB$$ is symmetric matrix if matrix multiplication of $$A$$ with $$B$$ is commutative.

The number of values of $$k$$ for which the linear equations
$$4x + ky + 2z = 0,kx + 4y + z = 0$$ and $$2x+2y+z=0$$ possess a non-zero solution is :

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 $$\left| A \right| = $$ determinant of matrix $$A$$.
Statement- 1: $$Tr$$$$(A)=0$$.
Statement- 2: $$\left| A \right| = 1$$ .