If $k<(\sqrt{2}+1)^3<k+2,$ where k is a natural number, the value of k?

If

 $\begin{bmatrix} x & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix} = \begin{bmatrix} 45 \end{bmatrix}$

then which one of the following is a value of x?

If

 $A = \begin{bmatrix} y & z & x \\ z & x & y \\ x & y & z \end{bmatrix} $

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. ∣M²∣=∣M∣²

II. $|M|=|M{}^{-1}|$

III. $|M|=|M{}^T|$

How many of the above are correct?