Minors, Cofactors, And Checkerboard Signs

From Deleted Submatrices to Cofactors

A determinant is built from products of matrix entries, with signs determined by permutations. Minors and cofactors organize those signed terms around a chosen position in the matrix. A minor is the determinant left after deleting a row and a column; the corresponding cofactor is that minor with an additional sign attached. Keeping these two quantities distinct is essential: the minor comes from a smaller matrix, while the cofactor also records the position of the deleted row and column.

Let \(\mathbf{A}=[a_{ij}]\) be an \(n\times n\) matrix over \(\mathbb{R}\), with \(n\ge 2\). To form a minor at position \((i,j)\), delete row \(i\) and column \(j\). The remaining \((n-1)\times(n-1)\) matrix is called the submatrix obtained by deleting row \(i\) and column \(j\). Its determinant is the \((i,j)\)-minor.

The sign depends on the position, not on whether the minor happens to be positive or negative. When \(i+j\) is even, the sign factor is \(+1\); when \(i+j\) is odd, it is \(-1\). Thus the sign factors, beginning in the upper-left corner, form the checkerboard

This pattern belongs to the signs used to turn minors into cofactors. It does not predict whether a minor itself is positive, negative, or zero. The determinant inside a minor must still be computed with its own determinant signs.

Why the Checkerboard Sign Appears

The factor \((-1)^{i+j}\) is not an arbitrary convention. It accounts for the sign change between a permutation of the original matrix that uses the entry \(a_{ij}\) and the corresponding permutation of the smaller matrix. The permutation formula for determinants gives a precise way to see this.

Proof. Write a term in the Leibniz formula using a permutation \(\sigma\) of \(\{1,\dots,n\}\): it is \(\operatorname{sgn}(\sigma)\prod_{r=1}^{n}a_{r,\sigma(r)}\). The permutations that use \(a_{ij}\) are exactly those with \(\sigma(i)=j\). Delete row \(i\) and column \(j\). The remaining assignments define a permutation \(\tau\) from the ordered remaining row positions to the ordered remaining column positions, which indexes a term in the Leibniz formula for \(M_{ij}\).

To compare the signs, count inversions: an inversion is a pair of positions \(r<s\) for which \(\sigma(r)>\sigma(s)\). Let \(x\) be the number of rows before \(i\) whose assigned columns are less than \(j\). Among the \(i-1\) rows before \(i\), the other \(i-1-x\) assignments exceed \(j\), giving \(i-1-x\) inversions involving row \(i\) on its left. Among the \(j-1\) columns less than \(j\), \(x\) have already been assigned to rows before \(i\). The other \(j-1-x\) are assigned to rows after \(i\), giving \(j-1-x\) further inversions involving row \(i\). The total number of inversions involving row \(i\) is therefore \(i+j-2-2x\), which has the same parity as \(i+j\). All other inversions are exactly the inversions of \(\tau\). Hence \(\operatorname{sgn}(\sigma)=(-1)^{i+j}\operatorname{sgn}(\tau)\). Summing the terms with \(\sigma(i)=j\) gives $$ a_{ij}(-1)^{i+j} \sum_{\tau}\operatorname{sgn}(\tau)\prod_{r\ne i}a_{r,\sigma(r)} =a_{ij}(-1)^{i+j}M_{ij} =a_{ij}C_{ij}. $$ Here the sum is the Leibniz formula for the determinant of the submatrix with row \(i\) and column \(j\) deleted. \(\square\)

This result concerns the group of permutation terms using one particular entry. It explains the signed minor without requiring the full determinant to be expanded along a row or column. The cofactor \(C_{ij}\) depends on the matrix entries outside row \(i\) and column \(j\); the entry \(a_{ij}\) appears separately in the grouped contribution \(a_{ij}C_{ij}\).

Proof. If \(i+j\) is even, \((-1)^{i+j}=+1\); if \(i+j\) is odd, \((-1)^{i+j}=-1\). Moving one position horizontally changes \(i+j\) by \(1\), and moving one position vertically also changes it by \(1\). Either move changes an even sum to an odd sum or an odd sum to an even sum, so the sign factor changes from \(+1\) to \(-1\) or from \(-1\) to \(+1\). Since \(1+1\) is even, the upper-left sign is \(+1\), and the alternating pattern follows across every row and down every column. \(\square\)

Computing Minors and Cofactors

Common Sign and Deletion Errors

A frequent error is to copy a minor directly into the cofactor matrix without checking its position. For example, in the three-by-three calculation above, a wrong calculation would be \(C_{23}=M_{23}=9\). But \(2+3=5\) is odd, so the correct calculation is \(C_{23}=(-1)^5M_{23}=-9\). Start the checkerboard with \(+\) at \((1,1)\), then alternate across each row and down each column; do not restart the pattern with \(+\) at the beginning of every row.

A different error is deleting a column with the wrong index. To find \(M_{23}\), delete row 2 and column 3—not row 2 and column 2. The remaining columns retain their original order. In a four-by-four matrix, deleting column 3 leaves columns 1, 2, and 4 in that order.

It is also important not to confuse the minor with the cofactor. The minor is already a determinant and may itself be negative. The checkerboard factor either keeps that value or reverses its sign; it does not replace the determinant calculation. When recording the results, a useful procedure is to write the entire matrix of minors first and apply the checkerboard in a separate step.

Minors and cofactors provide a compact way to keep determinant calculations organized. A table of minors records the smaller determinants; the checkerboard converts those values into cofactors. The permutation-sign argument explains why the position-dependent factor is necessary, while the examples show how to apply it without mixing the deletion step with the sign step.