List of practice Questions

Given:
\[ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{Bmatrix} x_1 \\ x_2 \\ x_3 \end{Bmatrix} = \begin{Bmatrix} 0 \\ 0 \end{Bmatrix} \]
The above system of equations represents a

The number of values of $k$ for which the following system of linear equations: \[ \begin{pmatrix} 4 & k & 2\\ k & 4 & 1\\ 2 & 2 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \] possess a non-zero solution, is _______

Consider the following system of equations:
$x - 2y + 3z = -1$
$x - 3y + 4z = 1$
$-2x + 4y - 6z = k$
The value of $k$ for which the system has infinitely many solutions is _______

Consider the following system of equations: \[ x - 2y + 3z = -1 \] \[ x - 3y + 4z = 1 \] \[ -2x + 4y - 6z = k \] The value of \( k \) for which the system has infinitely many solutions is \dots\dots.

For a system \(AX = b\) of linear equations to have infinitely many solutions, which one of the following options must be true?