Consistency of a System of linear equations

Question

Test the consistency of the system of linear equations

$$\begin{align} 4x-5y+z & =2 \\ 3x+y-2z& = 9 \\ x+4y+z& =5\end{align}$$

Answer 1

We have the system of linear equations given by: $$\begin{align} 4x-5y+z & =2\\ \\ 3x+y-2z& = 9 \\ \\ x+4y+z& =5\end{align}$$

This system can be represented by either of the following augmented coefficient matrices:

$$\begin{pmatrix} 4 & -5 & 1 &\mid 2 \\ 3 & 1 & -2 &\mid 9 \\ 1 & 4 & 1 &\mid 5\end{pmatrix}\to \quad \text{Swap Row 1 with Row 3}\to \quad \begin{pmatrix} 1 & 4 & 1 & \mid 5 \\ 3 & 1 & -2 & \mid 9 \\ 4 & -5 & 1 & \mid 2 \end{pmatrix}$$

On the second matrix, we can continue to perform elementary row operations to row reduce the matrix (i.e. using Gaussian Elimination) in order to check for consistency. Does this sound familiar?

If we row reduce to obtain a row, e.g., of $(0 \; 0 \; 0 \; c)$ where $c$ is non-zero, we have an inconsistent system. Else, the system is consistent. Consistency simplyy means: Can all equations simultaneously be true? If so, then the system is consistent. If not, then it is inconsistent. When a system is inconsistent, no solution can possibly exist. When it is consistent, either a unique solution exists, or infinitely many solutions exist.

If you row reduce correctly, to reduced row echelon form, you should obtain the matrix:

$$\begin{pmatrix} 1 & 0 & 0 &\mid 2\\ 0 & 1 & 0 & \mid 1 \\ 0 & 0 & 1 & \mid -1 \end{pmatrix}$$

Now, what can you say about the system's consistency?