Expansion Theorem for Determinants/Examples/Arbitrary Order 3 Example 1
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Example of Use of Expansion Theorem for Determinants
Consider the order $3$ determinant:
- $\begin {vmatrix} 1 & 2 & 3 \\ 4 & 1 & 2 \\ 6 & 5 & 4 \end {vmatrix}$
We have:
\(\ds \begin {vmatrix} 1 & 2 & 3 \\ 4 & 1 & 2 \\ 6 & 5 & 4 \end {vmatrix}\) | \(=\) | \(\ds 1 \times \begin {vmatrix} 1 & 2 \\ 5 & 4 \end {vmatrix} - 2 \times \begin {vmatrix} 4 & 2 \\ 6 & 4 \end {vmatrix} + 3 \times \begin {vmatrix} 4 & 1 \\ 6 & 5 \end {vmatrix}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \times \paren {1 \times 4 - 2 \times 5} - 2 \times \paren {4 \times 4 - 2 \times 6} + 3 \times \paren {4 \times 1 - 6 \times 5}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -6 - 8 + 42\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 28\) |
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): determinant
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): determinant