Matrices — NDA Mathematics PYQs

Consider the following statements :
I. If $n\times n$ ($n>1$) matrix is symmetric, then its inverse is also a symmetric matrix.
II. If $n\times n$ ($n>1$) matrix is singular, then its adjoint is also a singular matrix.
Which of the statements given above is/are correct ?

If $M=\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$, then what is the value of $|M||\text{adj}M|$ ?

If $M^T$ is the transpose of a $2\times 2$ matrix $M$, then which of the following is/are correct ?
I. $|M+M^T|=|M|+|M^T|$ if $M$ is symmetric.
II. $|M+M^T|=0$ if $M$ is anti-symmetric.
Select the answer using the code given below :

If $\left[\begin{array}{ccc}x& 1& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]\left[\begin{array}{c}1\\ 1\\ x\end{array}\right]=[45]$, then which one of the following is a value of $x$?

If $A=\left[\begin{array}{ccc}y& z& x\\ z& x& y\\ x& y& z\end{array}\right]$ where $x,y,z$ are integers, is an orthogonal matrix, then what is the value of $x^2+y^2+z^2$?

Consider the following in respect of a non-singular matrix $M$ :
I. $|M^2|=|M{|}^2$
II. $|M|=|M^{-1}|$
III. $|M|=|M^T|$
How many of the above are correct?

If $f(\theta )=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$, then what is $\{f(\pi ){\}}^2$ equal to?

If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$, then what is $A^2-4A$ equal to?

If $A=\left[\begin{array}{ccc}x& y& z\\ y& z& x\\ z& x& y\end{array}\right]$ where $x,y,z$ are integers, is an orthogonal matrix, then what is $A^2$ equal to?

What is the value of the determinant of the inverse of the matrix $\left[\begin{array}{cc}-4& -5\\ 2& 2\end{array}\right]$?