Let $A=\begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix}$ and $P=\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta > 0$. If $B=P A P^T$,$C=P^T B^{10} P$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$,where $\operatorname{gcd}(m, n)=1$,then $m+n$ is:

If ${a^2} + {b^2} + {c^2} = -2$ and $f(x) = \left| {\begin{array}{*{20}{c}}{1 + {a^2}x}&{(1 + {b^2})x}&{(1 + {c^2})x}\\{(1 + {a^2})x}&{1 + {b^2}x}&{(1 + {c^2})x}\\{(1 + {a^2})x}&{(1 + {b^2})x}&{1 + {c^2}x}\end{array}} \right|$,then $f(x)$ is a polynomial of degree:

Give the correct order of initials $T$ or $F$ for following statements. Use $T$ if statement is true and $F$ if it is false.
Statement $-1$ : If $A$ is an invertible $3 \times 3$ matrix and $B$ is a $3 \times 4$ matrix,then $A^{-1}B$ is defined.
Statement $-2$ : It is never true that $A + B, A - B$,and $AB$ are all defined.
Statement $-3$ : Every matrix none of whose entries are zero is invertible.
Statement $-4$ : Every invertible matrix is square and has no two rows the same.

Let $A$ be a $3 \times 3$ matrix such that $A^2 - 5A + 7I = 0$.
Statement-$I$: ${A^{-1}} = \frac{1}{7}(5I - A)$.
Statement-$II$: The polynomial $A^3 - 2A^2 - 3A + I$ can be reduced to $5(A - 4I)$.

If $A = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \{-1, 1\} \right\}$,then the number of singular matrices in $A$ is

If $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ and $\det(A^n - I) = 1 - \lambda^n$ for $n \in N$,then $\lambda$ is: