Simultaneous Linear Equations/Examples/Arbitrary System 6/Mistake
Source Work
1998: Richard Kaye and Robert Wilson: Linear Algebra:
- Part $\text I$: Matrices and vector spaces
- $1$ Matrices
- $1.5$ Row and column operations
- Solving linear equations
- $1.5$ Row and column operations
- $1$ Matrices
Mistake
To solve:
- $\begin {array} {rcrcrcr} x & + & y & + & 2 z & = & -1 \\ -x & + & & & z & = & -1 \\ -x & + & y & + & 4 z & = & 3 \\ \end {array}$,
first put the equation in matrix form
- $\paren {\begin {array} {rrr} 1 & 1 & 2 \\ -1 & 0 & 1 \\ -1 & 1 & 4 \end {array} } \begin {pmatrix} x \\ y \\ z \end {pmatrix} = \paren {\begin {array} {r} -1 \\ -1 \\ 3 \end {array} }$
and then put the augmented matrix formed from the matrix on the left with the column vector on the right into echelon form:
- $\paren {\begin {array} {rrr|r} 1 & 1 & 2 & -1 \\ -1 & 0 & 1 & -1 \\ -1 & 1 & 4 & 3 \end {array} } \to \paren {\begin {array} {rrr|r} 1 & 1 & 2 & -1 \\ 0 & 1 & 3 & -2 \\ 0 & 2 & 6 & -4 \end {array} } \to \paren {\begin {array} {rrr|r} 1 & 1 & 2 & -1 \\ 0 & 1 & 3 & -2 \\ 0 & 0 & 0 & 0 \end {array} }$.
Correction
That first transformation should be:
$\paren {\begin {array} {rrr|r} 1 & 1 & 2 & -1 \\ -1 & 0 & 1 & -1 \\ -1 & 1 & 4 & 3 \end {array} } \to \paren {\begin {array} {rrr|r} 1 & 1 & 2 & -1 \\ 0 & 1 & 3 & -2 \\ 0 & 2 & 6 & 2 \end {array} }$
which leads to:
- $\paren {\begin {array} {rrr|r} 1 & 1 & 2 & -1 \\ 0 & 1 & 3 & -2 \\ 0 & 0 & 0 & 6 \end {array} }$
Hence from the resulting $0 = 6$ it is seen that the initial system of simultaneous linear equations has no solutions.
The original system of simultaneous linear equations can be amended to:
- $\begin {array} {rcrcrcr} x & + & y & + & 2 z & = & -1 \\ -x & + & & & z & = & -1 \\ -x & + & y & + & 4 z & = & -3 \\ \end {array}$
from which the solution is derived in Simultaneous Linear Equations: Arbitrary System $6$.
$\blacksquare$
Sources
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations