Cofactor Expansion Along Any Row or Column

Expanding Beyond the First Row

The recursive definition of a determinant gives a way to expand along the first row. The same pattern works along every row and every column: each entry is multiplied by a signed determinant formed by deleting its row and column. This means a determinant can be calculated from whichever row or column is specified, although the choice may affect how much arithmetic is required.

We work with square matrices over \(\mathbb{R}\). For a matrix \(\mathbf{A}=[a_{ij}]\in\mathbb{R}^{n\times n}\), delete row \(i\) and column \(j\) to obtain an \((n-1)\times(n-1)\) matrix. Its determinant is the minor associated with entry \(a_{ij}\). The sign attached to that minor alternates in a checkerboard pattern.

The sign depends on the positions of the entry, not on the values of the matrix. The sign is positive when \(i+j\) is even and negative when \(i+j\) is odd. For example, the signs in a \(4\times4\) cofactor array are

In each sum, the cofactors belong to the same row or column as the entries being used. A row expansion uses \(a_{i1},\dots,a_{in}\) with \(C_{i1},\dots,C_{in}\); a column expansion uses \(a_{1j},\dots,a_{nj}\) with \(C_{1j},\dots,C_{nj}\). Cofactors are not reused from a different row or column.

Computing Along a Chosen Row

The check illustrates an important point: the cofactor values change when the expansion row changes, but the sum is still the same determinant. The theorem guarantees that equality. When verifying a computation, it is enough to recompute a second expansion correctly; the cofactors from the first expansion cannot simply be carried over.

Computing Along a Chosen Column

In both row and column expansions, signs belong to the positions \((i,j)\) of the entries in the original matrix. Deleting a row and column produces a smaller matrix for calculating the minor, but it does not reset the sign pattern. The sign remains \((-1)^{i+j}\) using the original indices.

One Matrix, Two Different Expansions

Consequences of the Expansion Formula

The cofactor expansion theorem also gives useful conclusions when entries in a row or column are zero. The results below are direct consequences; they do not require a separate determinant method.

Proof. For a row with only \(a_{ik}\) nonzero, expand along row \(i\): \(\det\mathbf{A}=\sum_{r=1}^{n}a_{ir}C_{ir}\). Every term with \(r\ne k\) is zero because \(a_{ir}=0\), leaving \(a_{ik}C_{ik}\). For a column with only \(a_{kj}\) nonzero, expand along column \(j\): \(\det\mathbf{A}=\sum_{r=1}^{n}a_{rj}C_{rj}\). Every term with \(r\ne k\) vanishes, leaving \(a_{kj}C_{kj}\). \(\square\)

Proof. If row \(i\) is zero, every entry \(a_{ik}\) in its cofactor expansion is zero, so \(\det\mathbf{A}=\sum_{k=1}^{n}0\cdot C_{ik}=0\). If column \(j\) is zero, its expansion gives \(\det\mathbf{A}=\sum_{k=1}^{n}0\cdot C_{kj}=0\). Thus either a zero row or a zero column forces the determinant to be zero. \(\square\)

These consequences explain why the expansion formula remains useful even when some entries vanish: zero terms contribute nothing, while each nonzero entry still requires its own cofactor. If a row or column has just one nonzero entry, the determinant reduces to that entry times one signed minor.

Common Sign and Indexing Pitfalls

Omitting the checkerboard sign gives a wrong cofactor. In the first worked example, the minor for \(a_{21}=3\) is \(\det\begin{bmatrix}2&0\\1&5\end{bmatrix}=10\). A wrong calculation sets \(C_{21}=10\), giving the row-2 contribution \(3(10)=30\). But \(2+1\) is odd, so \(C_{21}=(-1)^3(10)=-10\), and the correct contribution is \(3(-10)=-30\). The sign is part of the cofactor, even when the minor itself is positive.

Do not mix cofactors from different expansions. In the \(4\times4\) example, the row-4 cofactors are \((-2,1,-3,1)\), while the column-4 cofactors are \((-1,-1,-1,1)\). The row-4 entries must be paired with the first list; the column-4 entries must be paired with the second. Pairing entries from one row with cofactors from a different column does not follow the expansion formula.

Delete both the specified row and column. To calculate \(M_{ij}\), remove row \(i\) and column \(j\), then take the determinant of what remains. Removing only one of them produces a matrix of the wrong size and is not the required minor.

For every expansion, a reliable sequence is: fix the row or column; record its entries; calculate each minor by deleting the corresponding row and column; apply \((-1)^{i+j}\); then sum the entry-cofactor products. This keeps the indices, signs, and arithmetic separate enough to check.