Skill: Determine the sign of a cofactor from its position

Explanation and Free Practice Resources

In a square matrix, the sign attached to the cofactor at row $i$ , column $j$ is positive when $i+j$ is even and negative when $i+j$ is odd, forming an alternating checkerboard that begins with $+$ in the upper-left corner. This position-based sign is separate from the value of the minor and supplies the correct sign in determinant expansion.

On this page:

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

Detailed Explanation: Determine the sign of a cofactor from its position

For the cofactor in row $i$ , column $j$ , find the position sign using $i+j$ :

If $i+j$ is even, the sign is positive.
If $i+j$ is odd, the sign is negative.

Example: Determine the position sign of the cofactor in row $2$ , column $3$ of a matrix.

Identify the row and column: $i=2$ and $j=3$ .
Add them: $i+j=2+3=5$ .
Since $5$ is odd, the cofactor’s position sign is negative.

So the cofactor in position $(2,3)$ has sign $-$ . The sign depends only on the position, not on the value of the minor.

Learn by doing: Determine the sign of a cofactor from its position

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 - Cofactor Signs (3x3) - Matrix to Diagram

Earned ?

Download/Modify Free Grade 12 'Determine the sign of a cofactor from its position' Worksheet

Print this free 'Determine the sign of a cofactor from its position' 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 (Determine the sign of a cofactor from its position) through scaffolded problems that grow in difficulty as concepts are understood.