Expansion Theorem for Determinants/Examples

Examples of Use of Expansion Theorem for Determinants

Arbitrary Order $3$ Example

Consider the order $3$ determinant:

$\begin {vmatrix} 1 & 2 & 3 \\ 4 & 1 & 2 \\ 6 & 5 & 4 \end {vmatrix}$

We have:

\(\ds \begin {vmatrix} 1 & 2 & 3 \\ 4 & 1 & 2 \\ 6 & 5 & 4 \end {vmatrix}\)

\(=\)

\(\ds 1 \times \begin {vmatrix} 1 & 2 \\ 5 & 4 \end {vmatrix} - 2 \times \begin {vmatrix} 4 & 2 \\ 6 & 4 \end {vmatrix} + 3 \times \begin {vmatrix} 4 & 1 \\ 6 & 5 \end {vmatrix}\)

\(\ds \)

\(=\)

\(\ds 1 \times \paren {1 \times 4 - 2 \times 5} - 2 \times \paren {4 \times 4 - 2 \times 6} + 3 \times \paren {4 \times 1 - 6 \times 5}\)

\(\ds \)

\(=\)

\(\ds -6 - 8 + 42\)

\(\ds \)

\(=\)

\(\ds 28\)