If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ satisfies the equation $x^2 - (a + d)x + k = 0$,then

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & -\sin t & \cos t \end{bmatrix}$. Let $\lambda_{1}, \lambda_{2}, \lambda_{3}$ be the roots of $\det(A - \lambda I_{3}) = 0$, where $I_{3}$ denotes the identity matrix. If $\lambda_{1} + \lambda_{2} + \lambda_{3} = \sqrt{2} + 1$, then the set of possible values of $t$ for $-\pi \leq t < \pi$ is:

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 $A$ and $B$ are two matrices such that $AB = B$ and $BA = A$, then $A^{2} + B^{2}$ equals

If $a_{r} = \cos \frac{2 r \pi}{9} + i \sin \frac{2 r \pi}{9}$,$r = 1, 2, 3, \ldots$,$i = \sqrt{-1}$,then the determinant $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to:

If $M$ and $N$ are square matrices of order $3$, then which one of the following statements is not true?