Determinants Te...

If $$m = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}$$ and $$n = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}$$, then what is the value of the determinant of $$m \cos \theta - n \sin \theta$$?

Let $$A = \begin{pmatrix}x + 2 & 3x\\ 3 & x + 2\end{pmatrix}, B = \begin{pmatrix}x & 0\\ 5 & x + 2\end{pmatrix}$$. Then all solutions of the equation $$det (AB) = 0$$ is

$$\begin{vmatrix}1 & 1 & 1\\ a & b & c\\ a^2 - bc & b^2 - ca & c^2 - ab\end{vmatrix} = $$

If $$A = \begin{bmatrix} 1& 2\\ 3 & 4\end{bmatrix}$$, then $$A^{-1} =$$

If $$\alpha, \beta, \gamma$$ are the roots of the equation $$x^{3} + px + q = 0$$ then the value of the determinant $$\begin{vmatrix} \alpha& \beta & \gamma\\ \beta & \gamma & \alpha\\ \gamma & \alpha & \beta\end{vmatrix}$$ is

Consider the following statements in respect of the determinant $$\begin{vmatrix}{\cos}^2\dfrac{\alpha}{2}&{\sin}^2\dfrac{\alpha}{2}\\{\sin}^2\dfrac{\beta}{2}&{\cos}^2\dfrac{\beta}{2}\end{vmatrix}$$ where $$\alpha , \beta$$ are complementary angles
1. The value of the determinant is $$\dfrac{1}{\sqrt{2}}\cos \begin{pmatrix}\dfrac{\alpha - \beta}{2}\end{pmatrix}$$.
2. The maximum value of the determinant is $$\dfrac{1}{\sqrt{2}}$$.
Which of the above statements is/are correct?

If $$A = \begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3}\end{vmatrix}$$ and $$B = \begin{vmatrix} c_{1}& c_{2} & c_{3}\\ a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3}\end{vmatrix}$$ then.

If $$\begin{vmatrix}1 & sin \theta &1 \\ -sin \theta & 1 & sin \theta\\ -1 & -sin \theta & 1\end{vmatrix}$$ then,

If $$ C = 2 \cos \theta $$ , then the value of the determinant $$ \triangle = \begin{vmatrix} C & 1 & 0 \\ 1 & C & 1 \\ 6 & 1 & C  \end{vmatrix} $$ is :

If $$\begin{vmatrix} x & 2 & 8 \\ 2 & 8 & x \\ 8 & x & 2 \end{vmatrix}=\begin{vmatrix} 3 & x & 7 \\ x & 7 & 3 \\ 7 & 3 & x \end{vmatrix}=\begin{vmatrix} 5 & 5 & x \\ 5 & x & 5 \\ x & 5 & 5 \end{vmatrix}=0$$ then $$x$$ is equal to