The number of solutions of the equations $x + 4y - z = 0,$ $3x - 4y - z = 0,$ and $x - 3y + z = 0$ is

If $p, q, r$ are $3$ real numbers satisfying the matrix equation $[p, q, r] \begin{bmatrix} 3 & 4 & 1 \\ 3 & 2 & 3 \\ 2 & 0 & 2 \end{bmatrix} = [3, 0, 1]$,then $2p + q - r$ equals

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x + y + z = 1$,$x + \lambda y + z = 1$,and $x + y + \lambda z = 1$ is inconsistent. Then,$\sum_{\lambda \in S} (|\lambda|^2 + |\lambda|)$ is equal to

The solution of equations $x + y = 10$,$2x + y = 18$,and $4x - 3y = 26$ is:

The sum of three numbers is $6$. If we multiply the third number by $3$ and add the second number to it,we get $11$. By adding the first and third numbers,we get double the second number. Represent this algebraically and find the numbers using the matrix method.

The system of equations $x+y+z=5, x+2y+az=9, x+2y+z=b$ is inconsistent if