Let $A$ and $B$ be two $3 \times 3$ real matrices such that $(A^{2}-B^{2})$ is an invertible matrix. If $A^{5}=B^{5}$ and $A^{3} B^{2}=A^{2} B^{3}$,then the value of the determinant of the matrix $A^{3}+B^{3}$ is equal to:

Let $\alpha, \beta, \gamma, \delta$ be distinct imaginary roots of $z^5=1$. Find the value of the determinant: $\left| \begin{array}{ccc} e^{\alpha} & e^{2\alpha} & e^{3\alpha+1} \\ e^{\beta} & e^{2\beta} & e^{3\beta+1} \\ e^{\gamma} & e^{2\gamma} & e^{3\gamma+1} \end{array} \right|$.

The number of cubic polynomials $P(x)$ satisfying $P(1)=2, P(2)=4, P(3)=6, P(4)=8$ is

If $A = \begin{bmatrix} 2 & 3 \\ 0 & -1 \end{bmatrix}$,then the value of $\det(A^4) + \det(A^{10} - (\operatorname{adj}(2A))^{10})$ is equal to ........

Let $a$ and $b$ be non-zero real numbers such that $ab = 5/2$. Given $A = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ and $AA^T = 20I$ (where $I$ is the identity matrix),the quadratic equation whose roots are $a$ and $b$ is:

Let $A = \begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -5\alpha & 0 \\ 0 & 4\alpha & -2\alpha \end{bmatrix} + \text{adj}(A)$. If $\det(B) = 66$, then $\det(\text{adj}(A))$ equals: