Let $A$ be a matrix such that $AB$ is a scalar matrix where $B = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$ and $\det(3A) = 27$. Then $3A^{-1} + A^2 =$

Let $A$ be a $3 \times 3$ real matrix such that $A \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$,$A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}$,and $A \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}$. If $X = (x_1, x_2, x_3)^T$ and $I$ is an identity matrix of order $3$,then the system $(A - 2I)X = \begin{bmatrix} 4 \\ 1 \\ 1 \end{bmatrix}$ has:

If the system of equations $x+y+z=6$,$2x+5y+\alpha z=\beta$,and $x+2y+3z=14$ has infinitely many solutions,then $\alpha+\beta$ is equal to.

Give the correct order of initials $T$ or $F$ for following statements. Use $T$ if statement is true and $F$ if it is false.
Statement $-1$ : If the graphs of two linear equations in two variables are neither parallel nor the same,then there is a unique solution to the system.
Statement $-2$ : If the system of equations $ax + by = 0, cx + dy = 0$ has a non-zero solution,then it has infinitely many solutions.
Statement $-3$ : The system $x + y + z = 1, x = y, y = 1 + z$ is inconsistent.
Statement $-4$ : If two of the equations in a system of three linear equations are inconsistent,then the whole system is inconsistent.

The ordered pair $(a, b)$,for which the system of linear equations $3x - 2y + z = b$,$5x - 8y + 9z = 3$,and $2x + y + az = -1$ has no solution,is

The system of equations $3x + 2y + z = 6$, $3x + 4y + 3z = 14$ and $6x + 10y + 8z = a$ has an infinite number of solutions if $a$ is equal to