Read the following statements carefully:

A. Determinant is a square matrix

B. If A be any given square matrix of order n, then $A(\text{adj}A) = (\text{adj}A)A = |A|I$.

C. If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.

D. If A is a nonsingular matrix, then its inverse does not exist

Which of the above statements are true?

Choose the correct answer from the options given below:

For any invertible matrix $A$, which of the following is NOT true.

If matrix $A_p = \begin{bmatrix} p & p+1 \\ p-1 & p \end{bmatrix}_{p \in N}$ (where N is the set of natural numbers), then the value of $|A_1| + |A_2| + |A_3| + \dots + |A_{2025}|$ is:

If A is a skew-symmetric matrix, then which of the following statements is NOT true?

(A) A is singular if order of A is odd

(B) A is non-singular

(C) $A^{2025}$ is a skew-symmetric matrix

(D) $A^{2025}$ is a symmetric matrix

(E) all diagonal elements of A are zeros

Choose the correct answer from the options given below:

If P and Q are symmetric matrices, then QP-PQ is

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 :

If A is a singular matrix, then A{adj A} is equal to