Checkerboard Signs in Cofactors

From a Minor to a Signed Cofactor

When an entry of a matrix is paired with its minor, the row and column indices determine whether that minor keeps its sign or has its sign reversed. This adjustment is the cofactor sign. The rule depends only on the position of the entry, not on the values in the matrix, so the same alternating pattern applies to every square matrix of a given size.

Let \(\mathbf{A}\in\mathbb{R}^{n\times n}\), where \(n\geq 2\). Rows and columns are numbered starting at \(1\), so the upper-left entry is in position \((1,1)\). To form the minor associated with \(a_{ij}\), delete row \(i\) and column \(j\); the determinant of the resulting \((n-1)\times(n-1)\) matrix is \(M_{ij}\). Thus \(M_{ij}\) is a number, not the smaller matrix itself.

The sign is fixed by the position before the minor is calculated. For example, \(C_{12}=-M_{12}\) because \(1+2=3\) is odd, while \(C_{22}=M_{22}\) because \(2+2=4\) is even. These statements concern the sign factor only: if a minor is already negative, multiplying it by \(-1\) makes the cofactor positive.

Building the Checkerboard

Place \(+1\) in position \((1,1)\). Moving one step to the right or one step down changes the parity of \(i+j\), so the sign changes at every horizontal or vertical move. The resulting sign arrays begin as follows:

Each displayed \(+\) or \(-\) represents the sign factor multiplying the corresponding minor, not the value of the cofactor itself. For instance, a minus sign in the array tells us to calculate \(C_{ij}=-M_{ij}\); it does not guarantee that \(C_{ij}\) is a negative number.

Proof. By definition, the sign at \((i,j)\) is \((-1)^{i+j}\), which is \(+1\) for an even exponent and \(-1\) for an odd exponent. For a horizontal neighbor, the column index changes by \(1\) or \(-1\), while the row index stays fixed, so the sum of the indices changes by an odd number and its parity reverses. For a vertical neighbor, the row index changes by \(1\) or \(-1\), while the column index stays fixed; the sum again changes by an odd number, so the sign reverses. For a diagonal neighbor, both indices change by \(1\) or \(-1\). Their sum therefore changes by \(-2\), \(0\), or \(2\), each of which is even. The parity of the sum is unchanged, so the sign is unchanged. These neighbor statements apply only when the indicated neighboring position is inside the matrix. \(\square\)

This rule explains why the pattern alternates across every row and every column, while positions on the same diagonal direction have matching signs. The formula \((-1)^{i+j}\) is the reliable way to determine a sign; the checkerboard is a visual aid for reading that formula, not a separate calculation rule.

Calculating Cofactors in a Two-by-Two Matrix

Calculating Cofactors in a Three-by-Three Matrix

For a three-by-three matrix, deleting a row and column leaves a two-by-two matrix. Its determinant can be computed using \(ad-bc\) for \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\). Keep the two operations distinct: first calculate the minor determinant, then multiply by the position’s sign.

Proof. In a fixed row \(i\), the sign at column \(j\) is \((-1)^{i+j}\), and the sign at column \(j+1\) is \((-1)^{i+j+1}=-(-1)^{i+j}\). Thus consecutive signs in a row are opposite. In a fixed column \(j\), the sign at row \(i\) is \((-1)^{i+j}\), and the sign at row \(i+1\) is \((-1)^{i+j+1}=-(-1)^{i+j}\), so consecutive signs in a column are also opposite. For a diagonal-neighbor position, both indices change by \(1\) or \(-1\), so their sum changes by \(-2\), \(0\), or \(2\). Each change is even, leaving the parity of the sum, and hence the sign, unchanged. \(\square\)

Separating the Sign from the Minor

A cofactor can be positive, negative, or zero regardless of whether its sign factor is \(+1\) or \(-1\). The factor tells us only whether to preserve or reverse the minor’s value. In particular, if a minor is zero, multiplying by either sign leaves the cofactor zero. If the minor is negative and its sign factor is \(-1\), the cofactor is positive.

Common Sign and Indexing Errors

Mistake: assigning a negative sign to every minor in an off-diagonal position. In the first three-by-three example, position \((1,2)\) is off the main diagonal and its minor is \(-20\). A wrong calculation might write \(C_{12}=-20\), treating the negative sign factor as if it were the value of the cofactor. The correction is to multiply the sign factor by the minor: \(C_{12}=(-1)(-20)=20\). The sign factor reverses the minor’s sign; it does not replace the minor.

Mistake: starting the indices at zero. The position in the upper-left corner is \((1,1)\), not \((0,0)\). Its index sum is \(2\), so its sign is positive. Using zero-based indices can shift the parity and assign the wrong signs throughout the matrix. Begin counting rows and columns at \(1\), then use \((-1)^{i+j}\) for each entry.

Mistake: assuming a diagonal move always means down and right. A diagonal neighbor can be up-right, down-left, or up-left as well as down-right, provided it lies in the matrix. In every case both indices change by \(1\) or \(-1\), so the sum changes by an even number and the sign is preserved. The conclusion does not depend on a particular direction along the diagonal.

In the next step of determinant calculations, cofactors will be used together with the entries in their positions. For now, the essential procedure is specific: identify the row and column indices, read the sign from their parity, compute the minor, and multiply. Keeping those steps separate makes it easier to find and correct an error.