If $\begin{vmatrix} x & 3 & x \\ x^2 & x & 7 \\ x & x & 7 \end{vmatrix} = ax^4+bx^3+cx^2+dx+e$, then the value of $6a+5b+4c+3d+2e$ is

If A is a square matrix and I is the identity matrix of same order such that $A^2 = I$, then $3(A - I)^3 + 3(A + I)^3 - 15A$ is equal to

The matrix X such that it satisfies the equation: $X\begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$ is

Which of the following statements is (are) true?

(A) $B^T AB$ is a skew-symmetric matrix if A is a symmetric matrix

(B) $B^T AB$ is a symmetric matrix if A is a symmetric matrix

(C) $B^T AB$ is a symmetric matrix if A is a skew-symmetric matrix

(D) $B^T AB$ is a skew-symmetric matrix if B is a skew-symmetric matrix

(E) $B^T AB$ is a symmetric matrix if B is a symmetric matrix

Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 0 & 1 & -3 \\ -1 & 0 & 5 \\ 3 & -5 & 0 \end{bmatrix}$, then the value of $|A^{2025}|$ is

Assume A, B and C are matrices of order $m \times n$, $n \times 3$ and $3 \times q$ respectively. The restrictions on $_{m,n}$ and $_q$ so that $AB + BC$ is defined are

If the system of equation

$x - y + z = 4$

$x - 2y - 2z = 9$

$2x + y + \lambda z = 1$

has a unique solution, then