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.

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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.

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Matrices - Cofactor Signs (3x3) - Matrix to Diagram

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