The system of equations in 3 variables $x, y$ and $z$ as $x+y=5$, $y+az=3$, $x+by=8$ has a unique solution if the value of $a$ and $b$ are

(A) $a=1, b=1$

(B) $a=0, b=1$

(C) $a \in (-\infty,0)$ and $b\neq1$

(D) $a\neq0$ and $b\in(3,4)$

Choose the correct answer from the options given below:

If $A^2 - A + I = O$, where O is the zero matrix and I is the identity matrix, then $A^{-1}$ is

If $A=\begin{bmatrix}1 & 1 & 1\\1 & 0 & 3\\1 & -2 & 1\end{bmatrix}$, then

Match List-I with List-II

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If $A = \begin{bmatrix} 1 & 2 & -3 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 3 \\ 7 & -8 \\ 1 & 2 \end{bmatrix}$, then which of the following are true?

(A) AB is a $2 \times 3$ matrix

(B) $B^2$ is a $3 \times 3$ matrix

(C) BA is not possible

(D) $A^3$ is not possible

Choose the correct answer from the options given below:

The value/s of '$\lambda$' for which the system of equations

$\lambda x + 3y - z = 1$

$x + 2y + z = 2$

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

has unique solution is/are:

(where $R$ is set of real numbers)

The value of $\lambda$ for which the following system of equations has no solution is

$x+y+2z=4$

$2x+y+3z=5$

$x+3y+\lambda z=7$

The value of the determinant $\begin{vmatrix} a\cos\theta & b\sin\theta & 0 \\ -b\sin\theta & a\cos\theta & 0 \\ 0 & 0 & c \end{vmatrix}$ is :