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:

Let $A$ be a non-zero periodic matrix with period $4$ and $A^{12} + B = I$,where $I$ is the identity matrix and $B$ is any square matrix of the same order as $A$. The matrix product $AB$ is equal to:

Let $J_{n, m}=\int_{0}^{\frac{1}{2}} \frac{x^{n}}{x^{m}-1} d x, \quad \forall n>m$ and $n, m \in N$. Consider a matrix $A=\left[a_{i j}\right]_{3 \times 3}$ where $a_{i j}=J_{6+i, 3}-J_{i+3,3}$ for $i \leq j$ and $a_{i j}=0$ for $i>j$. Then $\left|\operatorname{adj} A^{-1}\right|$ is:

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$,then $\det[(I + B)^{50} - 50B]$ is equal to

In a $\Delta ABC,$ if $\left| \begin{array}{ccc} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{array} \right| = 0$,then $\sin^2 A + \sin^2 B + \sin^2 C = $

If $A(\theta)=\begin{bmatrix} i \sin \theta & \cos \theta \\ \cos \theta & i \sin \theta \end{bmatrix}$ is a matrix,where $i=\sqrt{-1}$,then which of the following is not true?