The number of all possible matrices of order $3 \times 3$ with each entry $-1, 0$ or $1$ 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

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If the matrix $A=\begin{bmatrix}0 & -1 & 3x\\1 & y & -5\\-6 & 5 & 0\end{bmatrix}$ is skew-symmetric, then the value of $(5x-y)$ is

If $A = [a_{ij}]_{3 \times 3}$ and $a_{ij} = i - j$, then which of the following statements are TRUE?

(A). $A$ is scalar matrix

(B). $A$ is skew symmetric matrix

(C). $|A| = 0$

(D). $a_{21} + a_{12} = 0$

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If A and B are symmetric matrices of the same order, then

Let $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ and $I$ be an identity matrix of order 3. If $A^TA=I$, then $x^2+y^2+z^2$ is equal to:

Which of the following are TRUE?

(A) The inverse of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is $\frac{1}{27}\begin{bmatrix} 7 & 3 \\ -2 & 3 \end{bmatrix}$.

(B) The determinant of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is 29.

(C) The inverse of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is $\frac{1}{29}\begin{bmatrix} 7 & 3 \\ -2 & 3 \end{bmatrix}$.

(D) The adjoint of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is $\begin{bmatrix} 7 & 3 \\ -2 & 3 \end{bmatrix}$.

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