Laplace's Expansion Theorem/Examples
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Examples of Laplace's Expansion Theorem
Arbitrary $3 \times 3$ Matrix
Let $\mathbf A$ be the matrix defined as:
- $\mathbf A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$
Then $\map \det {\mathbf A}$ can be calculated using Laplace's Expansion Theorem as follows.
Expanding row $2$:
\(\ds \map \det {\mathbf A}\) | \(=\) | \(\ds \paren {-1}^{2 + 1} \times 4 \begin {vmatrix} 2 & 3 \\ 8 & 9 \end {vmatrix}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {-1}^{2 + 2} \times 5 \begin {vmatrix} 1 & 3 \\ 7 & 9 \end {vmatrix}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {-1}^{2 + 3} \times 6 \begin {vmatrix} 1 & 2 \\ 7 & 8 \end {vmatrix}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds - 4 \paren {2 \times 9 - 3 \times 8}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 5 \paren {1 \times 9 - 3 \times 7}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds 6 \paren {1 \times 8 - 2 \times 7}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \paren {24 - 18} + 5 \paren {9 - 21} + 6 \paren {14 - 8}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
This shows that $\mathbf A$ is non-invertible.