Let $A$ be the set of all $3 \times 3$ determinants with entries $0$ or $1$ only and $B$ be the subset of $A$ consisting of all determinants with value $1$. If $C$ is the subset of $A$ consisting of all determinants with value $-1$, then:

Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix. Then the system of linear equations $(A^2 B^2 - B^2 A^2) X = 0$,where $X$ is a $3 \times 1$ column matrix of unknown variables and $0$ is a $3 \times 1$ null matrix,has:

Let $X = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \mathbb{R} \right\}$. If $f: X \rightarrow \mathbb{R}$ is defined by $f(A) = \det(A)$ for all $A \in X$, then $f$ is

$A$ is a $3 \times 3$ matrix satisfying $A^3-5A^2+7A+I=0$. If $A^5-6A^4+12A^3-6A^2+2A+2I=lA+mI$, then $l+m=$

If $A, B, C$ are the angles of a triangle,then the value of the determinant $\left| \begin{array}{ccc} \sin 2A & \sin C & \sin B \\ \sin C & \sin 2B & \sin A \\ \sin B & \sin A & \sin 2C \end{array} \right|$ is

If the polynomial $f(x) = \left|\begin{array}{ccc} (1+x)^{a} & (2+x)^{b} & 1 \\ 1 & (1+x)^{a} & (2+x)^{b} \\ (2+x)^{b} & 1 & (1+x)^{a} \end{array}\right|$, then the constant term of $f(x)$ is ($a$ and $b$ are positive integers).