Let $\mathrm{B}=\left[\begin{array}{ccc}5& 0& 0\\ 2& -1& 0\\ 4& 6& 2\end{array}\right]$ . Then $\det ({\mathrm{B}}^2-2\mathrm{I})=$ _____.

For the matrix $\mathrm{A}=\left[\begin{array}{cc}4& 2\\ 2& 4\end{array}\right]$ , the eigen value corresponding to the eigen vector $\left[\begin{array}{c}101\\ 101\end{array}\right]$ is _____.

Let $\mathrm{P}=\left[\begin{array}{ccc}-1& 0& 2\\ 0& 0& 5\\ 0& 0& 10\end{array}\right]$ . Then the sum of eigen values of ${\mathrm{P}}^2$ is _____.

If $\mathrm{A}=\left[\begin{array}{cc}1& 2\\ -1& 3\end{array}\right]$ . Then ${\mathrm{A}}^5-3{\mathrm{A}}^4+{\mathrm{A}}^3-7{\mathrm{A}}^2+5\mathrm{A}+\mathrm{I}$ is equals to _____.

Let $\mathrm{A}$ be a matrix of order $6\times 6$ such that $\mathrm{A}.\mathrm{M}.\left(2\right)=3$, $\mathrm{A}.\mathrm{M}.\left(3\right)=2$ and $\mathrm{A}.\mathrm{M}.\left(5\right)=1$, then the product of eigen values of $\mathrm{A}$ is _____.

Which of the following matrices is in row echelon form?

$\mathrm{\rho }\left({\mathrm{I}}_4\right)=$ _____, where ${\mathrm{I}}_4$ is an identity matrix of order 4 .

The system of simultaneous linear equations $\mathrm{x}+2\mathrm{y}+5\mathrm{z}=5, 2\mathrm{x}+4\mathrm{y}+6\mathrm{z}=10, 3\mathrm{x}+2\mathrm{y}+\mathrm{z}=3$ has _____.

Let $\mathrm{A}$ be a non-invertible matrix of order 4 . If eigen values of $\mathrm{A}$ are 2, 3 and -4 , then the other eigen value of $\mathrm{A}$ is _____.

Let $\mathrm{A}=\left[\begin{array}{ccc}1& 2& 4\\ 3& 0& 6\\ 1& 1& \mathrm{k}\end{array}\right]$ has one eigen value equals to 3 . Then sum of the other two eigen value is _____.