Which of the following is(are) NOT the square of a $3\times 3$ matrix with real entries?

Let $A$ be a $2\times 2$ matrix

Statement $1:$ $adj\left(adjA\right)=A$

Statement $2:$ $\left|adjA\right|=|A\mid$.

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. $Tr\left(A\right)=$ sum of diagonal elements of $A$ and $\left|\begin{array}{c}A\end{array}\right|$ is determinant of matrix $A.$

Statement-$1$: $Tr\left(A\right)=0$

Statement-$2$: $\left|\begin{array}{c}A\end{array}\right|=1$

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

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

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

$4x+ky+2z=0$
$kx+4y+z=0$
$2x+2y+z=0$

posses a non-zero solution is