Skill: Expand a determinant along a specified column

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For a square matrix of manageable order, expansion along a specified column expresses its determinant as the sum of each entry in that column multiplied by its cofactor: the signed determinant of the smaller matrix formed by deleting that entry’s row and its column. The alternating cofactor signs depend on the entry’s position, not its value; the focus is numerical determinant evaluation rather than arbitrary-order theory or abstract generalizations.

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Detailed Explanation: Expand a determinant along a specified column

To expand a determinant along a column, use each entry in that column together with the determinant of the smaller matrix left after deleting that entry’s row and its column. The cofactor signs alternate; for a $3 \times 3$ matrix, the signs in the second column are $-,+,-$ .

For example, expand along the second column:

\det\begin{pmatrix} 2&1&3\\ 0&4&-1\\ 5&2&2 \end{pmatrix}

For each entry in the second column, delete its row and the second column to get its $2 \times 2$ minor:

\begin{aligned} \det &=-1\begin{vmatrix}0&-1\\5&2\end{vmatrix} +4\begin{vmatrix}2&3\\5&2\end{vmatrix} -2\begin{vmatrix}2&3\\0&-1\end{vmatrix}\\ &=-1(0\cdot2-(-1)\cdot5) +4(2\cdot2-3\cdot5) -2(2\cdot(-1)-3\cdot0)\\ &=-5-44+4\\ &=-45. \end{aligned}

The signs $-,+,-$ come from the entries’ positions, not from whether the entries themselves are positive or negative.

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Matrices - Determinant (3x3) Cofactor Expansion - Matrix to Answer

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