Let
\[ A = \left[ \begin{array}{c c c} 3 & - 1 & 2 \\ 2 & 1 & 1 \\ 1 & - 3 & 0 \end{array} \right]. \]
For which triples \((y_{1}, y_{2}, y_{3})\) does the system \(AX = Y\) have a solution?
Solution
The system \(AX=Y\) has a solution for every triple \[ (y_1,y_2,y_3)\in F^3. \]
1. The coefficient matrix \(A\) is invertible.
Its determinant is \[ \det\begin{bmatrix} 3&-1&2\\ 2&1&1\\ 1&-3&0 \end{bmatrix}=-6\ne0. \]
2. Hence \(AX=Y\) is solvable for every \(Y\in F^3\).
By 1, \(A^{-1}\) exists. For arbitrary \(Y=(y_1,y_2,y_3)^t\), take \[ X=A^{-1}Y. \] Then \(AX=Y\). In fact the solution is unique.