Skill: Expand a determinant along a specified column

Explanation and Free Practice Resources

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.

On this page:

Learn “Expand a determinant along a specified column” with a detailed explanation of the concept
Master “Expand a determinant along a specified column” with unlimited, free practice problems
Download (or modify) a free worksheet on Expand a determinant along a specified column you can use at home or in class with practice problems that scaffold and explore “Expand a determinant along a specified column” from different perspectives.
Start working on adaptive units like “Matrix Operations - Determinants, Co-factors, Adjoints” or “Matrix Operations - Transpose, Determinants” that introduce, teach, or practice “Expand a determinant along a specified column” to maximize your learning.

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.

Learn by doing: Expand a determinant along a specified column

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Your results

Practice with unlimited practice problems

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

Earned ?

Download/Modify Free Grade 12 'Expand a determinant along a specified column' Worksheet

Print this free 'Expand a determinant along a specified column' worksheet to practice this skill away from the computer. Click 'Modify' to easily customize this worksheet to your exact needs.

Units Teaching This Skill

These adaptive units introduce, teach, or reinforce this skill (Expand a determinant along a specified column) through scaffolded problems that grow in difficulty as concepts are understood.