$A$ determinant is chosen at random from the set of all determinants of order $2$ with elements $0$ or $1$ only. The probability that the value of the chosen determinant is positive is:

For $A = \begin{bmatrix} 0 & 0 & -2 \\ 0 & -2 & 0 \\ -2 & 0 & 0 \end{bmatrix}$,which of the following is true?

If the matrix $M_r$ is given by $M_r = \begin{bmatrix} r & r-1 \\ r-1 & r \end{bmatrix}$ for $r = 1, 2, 3, \ldots$,then $\det(M_1) + \det(M_2) + \ldots + \det(M_{2008}) = $

For $i=1, 2, 3$ and $j=1, 2, 3$. If $a_i^2+b_i^2+c_i^2=1$,$a_i a_j+b_i b_j+c_i c_j=0$,$\forall i \neq j$ and $A=\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}$,then $\det(AA^T)=$

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

The equation $\left| \begin{array}{ccc} (1+x)^2 & (1-x)^2 & -(2+x^2) \\ 2x+1 & 3x & 1-5x \\ x+1 & 2x & 2-3x \end{array} \right| + \left| \begin{array}{ccc} (1+x)^2 & 2x+1 & x+1 \\ (1-x)^2 & 3x & 2x \\ 1-2x & 3x-2 & 2x-3 \end{array} \right| = 0$