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 matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 4 \\ 0 & 3 & 2 \end{bmatrix}$ and $A^3 - 8A^2 + \alpha A + \beta I = O$,then the ordered pair $(\alpha, \beta)$ is

If the function $f:[a, b] \rightarrow \left[-\frac{\sqrt{3}}{4}, \frac{1}{2}\right]$ defined by $f(x) = \left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+\sin x & 1 \\ 1+\cos x & 1 & 1 \end{array} \right|$ is one-one and onto, then:

Let $B=\begin{bmatrix} 2 & 6 & 4 \\ 1 & 0 & 1 \\ -1 & 1 & -1 \end{bmatrix}$ and $C=\begin{bmatrix} -1 & 0 & 1 \\ 1 & 1 & 3 \\ 2 & 0 & 2 \end{bmatrix}$. If a matrix $A$ is such that $BAC=I$,then $A^{-1}=$

$A$ and $B$ are $3 \times 3$ matrices such that $AB + A + B = 0$,then:

Let $A = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$ and $B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$ be two $2 \times 1$ matrices with real entries such that $A = XB$,where $X = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & -1 \\ 1 & k \end{bmatrix}$ and $k \in R$. If $a_1^2 + a_2^2 = \frac{2}{3}(b_1^2 + b_2^2)$ and $(k^2 + 1)b_2^2 \neq -2b_1b_2$,then the value of $k$ is ....... .