$\Delta=\left|\begin{array}{ccc}1 & \cos x & 1 \\ -\cos x & 1 & \cos x \\ -1 & -\cos x & 1\end{array}\right|$ (A) $\Delta=2\left(1-\cos ^{2} x\right)$ (B) $\Delta=2\left(2-\sin ^{2} x\right)$ (C) Minimum value of $\Delta$ is $2$ (D) Maximum value of $\Delta$ is $4$ Choose the correct answer from the options given below :

The area of a triangle whose vertices are $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$ is given by the absolute value of

If $A=\begin{bmatrix}1 & 2\\2 & 1\end{bmatrix}$ and $f(x)=x^2-2x-3$, then $f(A)$ is equal to

Which of the following statement is/are correct?

(A) A square matrix $A = [a_{ij}]$ is called a symmetric matrix if $a_{ij} = a_{ji}$ for all $i, j$

(B) $A = [a_{ij}]_{m \times m}$ is a diagonal matrix if $a_{ij} = 0$ when $i = j$

(C) A square matrix $A = [a_{ij}]$ is called a skew symmetric matrix, if $a_{ij} = -a_{ji}$ for all $i, j$

(D) The multiplication of diagonal matrices of same order is commutative

Choose the correct answer from the options given below:

Let $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -6 \\ -2 & 4 \end{bmatrix}$

(A) $\det(A^T) = 1$

(B) $AB = I$, where $I$ is the identity matrix of order 2.

(C) $A^{-1} = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$

(D) adj $(B) = \begin{bmatrix} 4 & 2 \\ 6 & 4 \end{bmatrix}$

Choose the correct answer from the options given below:

If $P = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}$, $Q = \begin{bmatrix} 4 & 0 \\ 1 & 5 \end{bmatrix}$ and $R = \begin{bmatrix} 2 & 0 \\ 1 & -2 \end{bmatrix}$, then which of the following are TRUE?

(A) $(P^T)^T = P^T$

(B) $(P-Q)R = PR - QR$

(C) $(P-Q)^T = Q^T - P^T$

(D) $(PQ)^T = Q^TP^T$

Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then $A^{-1}$ is equal to