Common Sign and Minor Errors

Three Places a Cofactor Calculation Can Go Wrong

A cofactor expansion is a sum of contributions, one for each entry in a chosen row or column. Each contribution depends on three pieces of information: the entry, the determinant left after deleting its row and column, and a checkerboard sign. A correct calculation keeps all three pieces attached to the same position. Many errors arise when one piece is copied from a neighbouring position or when the unsigned minor is mistaken for the signed cofactor.

Throughout, let \(\mathbf{A}=[a_{ij}]\) be an \(n\times n\) real matrix with \(n\ge 2\). For an entry in row \(i\), column \(j\), deleting a row means removing all its entries, and deleting a column means removing all its entries. The remaining square matrix has \(n-1\) rows and \(n-1\) columns. Its determinant is the minor associated with that entry.

The factor \((-1)^{i+j}\) gives the checkerboard pattern. Starting with a positive sign in position \((1,1)\), signs alternate across each row and down each column:

For instance, the sign in position \((2,3)\) is positive because \(2+3=5\) is odd? No: the sign is negative when the sum is odd, so position \((2,3)\) has a negative sign. In general, \(i+j\) even gives a positive sign, and \(i+j\) odd gives a negative sign. The checkerboard is attached to positions in the original matrix, not to the entries’ values.

Keeping the Expansion Indices Together

The cofactor expansion along row \(r\) is \(\det\mathbf{A}=\sum_{j=1}^{n}a_{rj}C_{rj}\). Along column \(s\), it is \(\det\mathbf{A}=\sum_{i=1}^{n}a_{is}C_{is}\). These are the cofactor expansion formulas from the earlier material on determinants by cofactor expansion. The bookkeeping is the same in either direction: retain the entry’s original position, find the minor by deleting the corresponding row and column, and multiply that minor by its position’s sign.

When expanding along row \(r\), the row deleted from each minor is always row \(r\). The column deleted changes with the term: the term involving \(a_{rj}\) uses the minor \(M_{rj}\), made by deleting column \(j\). For a column expansion, the column deleted is fixed and the deleted row changes. This distinction is easy to lose when writing several small determinants in sequence.

Worked Examples: Signs, Deletions, and Cofactors

What a Single Mistake Does to the Answer

The size of a discrepancy can help identify its source. The following two results assume a single error in one term and that every other contribution is kept unchanged. They do not say that every incorrect calculation must produce a different number: a zero entry or a coincidental equality can conceal an error.

Proof. The correct term is \(a_{rk}C_{rk}\). Reversing only its cofactor sign replaces it with \(a_{rk}(-C_{rk})=-a_{rk}C_{rk}\). All other terms cancel when we subtract the correct sum from the incorrect sum. Therefore \(D_{\mathrm{wrong}}-D=-a_{rk}C_{rk}-a_{rk}C_{rk}=-2a_{rk}C_{rk}\). If \(a_{rk}=0\) or \(C_{rk}=0\), the difference is zero, so this particular sign error may not change the numerical answer. \(\square\)

In the first worked example, \(a_{23}=4\) and \(C_{23}=5\). The proposition predicts a change of \(-2(4)(5)=-40\). Indeed, \(-45-(-5)=-40\). This gives a useful arithmetic check when the incorrect and correct totals are both available.

Proof. The correct term is \(a_{rk}(-1)^{r+k}M_{rk}\). The substituted term is \(a_{rk}(-1)^{r+k}M_{st}\), because the sign still belongs to the original position \((r,k)\). Subtracting the correct term from the substituted term gives \(a_{rk}(-1)^{r+k}(M_{st}-M_{rk})\). Every other term is unchanged and cancels in the difference of the two sums. \(\square\)

In the second worked example, \(a_{21}=3\), the position sign is negative, the correct minor is \(10\), and the mistakenly used minor is \(9\). The predicted change is \(3(-1)(9-10)=3\). The incorrect result exceeds the correct one by \(-2-(-5)=3\), as predicted.

A Practical Check Before Adding

Before summing an expansion, check each term in isolation. This is more reliable than trying to recover a lost sign after combining several positive and negative numbers. In particular, do not infer a cofactor’s sign from the minor’s value: a negative minor remains negative if its position sign is positive, but becomes positive if its position sign is negative.

• Sign: Calculate \((-1)^{i+j}\) from the entry’s position in the original matrix.
• Deletion: Cross out row \(i\) and column \(j\) for the term \(a_{ij}\).
• Minor: Evaluate the determinant of what remains before applying the cofactor sign.
• Contribution: Multiply the original entry by the signed cofactor, not by the unsigned minor.
• Check: If practical, expand along a different row or column and compare the result.

A different expansion is a check on the calculation, not permission to change the determinant. Each choice of row or column must yield the same value when its entries, minors, and positional signs are all handled correctly. If two expansions disagree, compare their terms one at a time; the error is often visible in a single deleted row, deleted column, or checkerboard sign.

Common Pitfall: The Sign Belongs to the Position

A tempting shortcut is to put a minus sign on a term whenever its minor is negative. That confuses the minor’s numerical sign with the cofactor’s positional sign. For the middle term in the third example, \(M_{12}=5\) is positive, but the position \((1,2)\) has a negative checkerboard sign, so \(C_{12}=-5\). The contribution is \(1(-5)\), not \(1(5)\). Conversely, a negative minor in a positive-sign position remains a negative cofactor, as \(M_{13}=-20\) showed.

Another risky shortcut is to write several \(2\times2\) determinants without labeling the positions that produced them. A determinant may be evaluated correctly and still be the wrong minor for the term. Keeping an explicit label such as \(M_{21}\) beside each small determinant makes that error easier to catch. The row and column indices are part of the definition, not optional bookkeeping.