Simultaneous Linear Equations/Examples/Arbitrary System 9

From ProofWiki
Jump to navigation Jump to search

Example of Simultaneous Linear Equations

Let $S$ denote the system of simultaneous linear equations:

\(\ds x + 2 y + z\) \(=\) \(\ds 1\)
\(\ds x + y + z\) \(=\) \(\ds 1\)
\(\ds -x + z\) \(=\) \(\ds 1\)


$S$ has the single solution:

\(\ds x\) \(=\) \(\ds 0\)
\(\ds y\) \(=\) \(\ds 0\)
\(\ds z\) \(=\) \(\ds 1\)


Proof

We express $S$ in matrix representation:

$\begin {pmatrix} 1 & 2 & 1 \\ 1 & 1 & 1 \\ -1 & 0 & 1 \end {pmatrix} \begin {pmatrix} x \\ y \\ z \end {pmatrix} = \begin {pmatrix} 1 \\ 1 \\ 1 \end {pmatrix}$

and consider the augmented matrix:

$\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix} = \paren {\begin {array} {ccc|c} 1 & 2 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ -1 & 0 & 1 & 1 \end {array} }$


In the following, $\sequence {e_n}_{n \mathop \ge 1}$ denotes the sequence of elementary row operations that are to be applied to $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$.

The matrix that results from having applied $e_1$ to $e_k$ in order is denoted $\begin {pmatrix} \mathbf A_k & \mathbf b_k \end {pmatrix}$.


$e_1 := r_2 \to r_2 - r_1$

$e_2 := r_3 \to r_3 + r_1$

Hence:

$\begin {pmatrix} \mathbf A_2 & \mathbf b_2 \end {pmatrix} = \paren {\begin {array} {ccc|c} 1 & 2 & 1 & 1 \\ 0 & -1 & 0 & 0 \\ 0 & 2 & 2 & 2 \\ \end {array} }$


$e_3 := r_1 \to r_1 + 2 r_2$

$e_4 := r_3 \to r_3 + 2 r_2$

Hence:

$\begin {pmatrix} \mathbf A_4 & \mathbf b_4 \end {pmatrix} = \paren {\begin {array} {ccc|c} 1 & 0 & 1 & 1 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 2 \\ \end {array} }$


$e_5 := r_2 \to -r_2$

$e_6 := r_3 \to \dfrac 1 2 r_3$

Hence:

$\begin {pmatrix} \mathbf A_6 & \mathbf b_6 \end {pmatrix} = \paren {\begin {array} {ccc|c} 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ \end {array} }$


$e_7 := r_1 \to r_1 - r_3$

Hence:

$\begin {pmatrix} \mathbf A_7 & \mathbf b_7 \end {pmatrix} = \paren {\begin {array} {ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ \end {array} }$


Thus:

\(\ds x\) \(=\) \(\ds 0\)
\(\ds y\) \(=\) \(\ds 0\)
\(\ds z\) \(=\) \(\ds 1\)

$\blacksquare$


Sources