Matrices — NDA Mathematics PYQs

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?

Consider the following in respect of non-singular matrices $A$ and $B$ :
I. $(AB{)}^{-1}=A^{-1}{B}^{-1}$
II. $(BA)(AB{)}^{-1}=I$, where $I$ is the identity matrix
III. $(AB{)}^T=A^T{B}^T$
How many of the above are correct?

Consider the following for the two (02) items that follow :
Let $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$.

What is $(\text{adj}A{)}^{-1}$ equal to?

Consider the following statements :
Statement-I :
If $X$ is an $n\times n$ matrix, then $\det (mX)=m^n\det (X)$, where $m$ is a scalar.
Statement-II :
If $Y$ is a matrix obtained from $X$ by multiplying any row or column by a scalar $m$, then $\det (Y)=m\det (X)$.
Which one of the following is correct in respect of the above statements?

Consider the following statements about the matrix $M=\left[\begin{array}{ccc}71& 23& 48\\ 57& 28& 29\\ 65& 17& 48\end{array}\right]$ :
Statement-I : The inverse of $M$ does not exist.
Statement-II : $M$ is non-singular.
Which one of the following is correct in respect of the above statements?