If a, b and c are all different from zero and $\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix} = 0$, then the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is :

If $A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, B = \begin{bmatrix} 0 & 0 \\ 3 & 0 \end{bmatrix}$ then

The equations in terms of $x$ and $y$ are:

If A is square matrix of order 3 x 3 such that $a_{ij} = j - i$, then which of the following statements are correct?

(A) A is a symmetric matrix.

(B) A is a skew symmetric matrix.

(C) $|A| = 0$.

(D) A is a diagonal matrix.

Choose the correct answer from the options given below:

The system of equation $x-y+z=4, x-2y-2z=9, 2x+y+\alpha z=1$ has unique solution then value(/s) of $\alpha$ is (/are) (where $\mathbb{R}$ is set of real numbers)

If $A = \begin{bmatrix} 0 & x & x \\ 2y & y & -y \\ z & -z & z \end{bmatrix}$ such that $AA^T=2I$ (where I is an identity matrix of order 3), then which of the following are True?

A. $x^2=1, 3y^2=2$

B. $6y^2=4, 2z^2=3$

C. $9z^2=6, x^2=1$

D. $x^2:y^2:z^2 = 3:1:2$

Choose the correct answer from the options given below:

If A and B are invertible matrices of order 3, $|A| = 2$ and $|(AB)^{-1}| = -\frac{1}{6}$, then the value of $|B|$ is :