If each element of a second order determinant is either zero or one,what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently,each value being assumed with probability $\frac{1}{2}$).

Let $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $P = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$. Let $Q = \begin{bmatrix} x & y \\ z & 4 \end{bmatrix}$ for some non-zero real numbers $x, y$,and $z$,for which there exists a $2 \times 2$ matrix $R$ with all entries being non-zero real numbers,such that $QR = RP$. Then which of the following statements is (are) true?
$(A)$ $|Q-2I| = 0$
$(B)$ $|Q-6I| = 12$
$(C)$ $|Q-3I| = 15$
$(D)$ $yz = 2$

Let $A$ be a symmetric matrix and $B$ be a skew-symmetric matrix,such that $A - B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Then $|A|$ is:

Let $A = [a_{ij}]_{2 \times 2}$ where $a_{ij} \neq 0$ for all $i, j$ and $A^2 = I$. Let $a$ be the sum of all diagonal elements of $A$ and $b = |A|$. Then $3a^2 + 4b^2$ is equal to:

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

Let $\theta = \frac{\pi}{5}$ and $A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$. If $B = A + A^4$,then $\det(B)$