Let $A = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 10 & 20 & 30 \\ 11 & 22 & k \end{array}\right]$ and $\mathbf{v} = \left[\begin{array}{l} x \\ y \\ z \end{array}\right]$, where $k$ is a constant and $x, y, z$ are variables. Statements (9) Regardless of the value of $k$, the matrix $A$ is not invertible, i.e., there is no $3 \times 3$ matrix $B$ such that $BA =$ the $3 \times 3$ identity matrix. (10) There is a unique $k$ such that determinant of $A$ is 0. (11) The set of solutions $(x, y, z)$ of the matrix equation $A\mathbf{v} = \left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right]$ is either a line or a plane containing the origin. (12) If the equation $A\mathbf{v} = \left[\begin{array}{c} p \\ q \\ r \end{array}\right]$ has a solution, then it must be true that $q = 10p$.
Let $A = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 10 & 20 & 30 \\ 11 & 22 & k \end{array}\right]$ and $\mathbf{v} = \left[\begin{array}{l} x \\ y \\ z \end{array}\right]$, where $k$ is a constant and $x, y, z$ are variables.
Statements
(9) Regardless of the value of $k$, the matrix $A$ is not invertible, i.e., there is no $3 \times 3$ matrix $B$ such that $BA =$ the $3 \times 3$ identity matrix.\\
(10) There is a unique $k$ such that determinant of $A$ is 0.\\
(11) The set of solutions $(x, y, z)$ of the matrix equation $A\mathbf{v} = \left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right]$ is either a line or a plane containing the origin.\\
(12) If the equation $A\mathbf{v} = \left[\begin{array}{c} p \\ q \\ r \end{array}\right]$ has a solution, then it must be true that $q = 10p$.