If $A=\begin{bmatrix}a & 1 & c\\ b & 2 & 1\\ 2 & 1 & c\end{bmatrix}$ is a singular matrix, then

Let $A=\begin{bmatrix}1 & \sin\theta & 1\\-\sin\theta & 1 & \sin\theta\\-1 & -\sin\theta & 1\end{bmatrix}$, where $\theta\in[0,2\pi]$ then which of the following statements are correct?

A. $|A|=2+2\sin^2\theta$

B. Maximum value of $|A|$ is 1.

C. Maximum value of $|A|$ is –1.

D. $|A|\in[2,4]$

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Match List-I with List-II

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Which of the following statements are correct?

(A) If A is a square matrix, then $|A^2| = |A|^2$.

(B) If A and B are square matrices of the same order, then det (AB) = det (A) + det (B).

(C) If A is a square matrix of order 3 and $|A|=2$, then the value of $|-3A|$ is 54.

(D) If the matrix $\begin{bmatrix} 5 -x & x -1 \\ 3 &5 \end{bmatrix}$ is singular, then the value of x is 7/2.

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If $3A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y \end{bmatrix}$ and $AA^T = I$, then $x + y$ is equal to

For any 3 ×3 matrix $A$, if $A(adj\,A) = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$, then $2|A|$ is equal to

The set of values of K for which the system of equations $\begin{bmatrix} 2 & 3 & 1 \\ 4 & 5 & 0 \\ 1 & K & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$ gives a unique solution is :