A Cofactor Expansion Problem-Solving Workshop

A Reliable Plan for Cofactor Expansion

Cofactor expansion turns a determinant into a sum of smaller determinants. The arithmetic is usually manageable; the challenge is organising it so that signs, deleted rows and columns, and smaller determinants are all handled correctly. A useful plan is to choose an expansion direction before calculating, write down exactly which rows and columns each minor uses, and check the result by a second expansion when practical.

For an \(n\times n\) matrix \(\mathbf{A}=[a_{ij}]\), the minor \(M_{ij}\) is the determinant of the \((n-1)\times(n-1)\) matrix obtained by deleting row \(i\) and column \(j\). The cofactor is \(C_{ij}=(-1)^{i+j}M_{ij}\). Thus an entry contributes its value times its cofactor, not merely its minor. The alternating signs follow this checkerboard pattern:

By the cofactor expansion theorem, expanding along row \(i\) gives \(\det\mathbf{A}=\sum_{j=1}^{n}a_{ij}(-1)^{i+j}M_{ij}\); expanding along column \(j\) gives \(\det\mathbf{A}=\sum_{i=1}^{n}a_{ij}(-1)^{i+j}M_{ij}\). This theorem is established in the earlier material on cofactor expansion. A zero entry contributes zero, so an expansion direction with many zero entries requires fewer minor calculations.

Proof. The cofactor expansion theorem says that the determinant is the sum of the terms \(a_{ij}(-1)^{i+j}M_{ij}\) along the selected row or column. For every zero entry, its term is \(0\cdot(-1)^{i+j}M_{ij}=0\), so removing those terms does not change the sum. If exactly one entry is nonzero, every other term is zero and the sum consists of that one term. \(\square\)

The rule is simple, but a minor must be formed from the original matrix by deleting the correct row and column. It is not the determinant of the remaining entries in their original locations, and it does not include the sign factor. Keep the minor and its cofactor sign separate until the final multiplication.

First-Row Expansion of a Three-by-Three Matrix

For a \(3\times3\) matrix, each minor in a cofactor expansion is \(2\times2\). Calculate these smaller determinants using \(ad-bc\), and show both products before subtracting. That makes sign errors easier to locate: a negative result for a minor is not itself the cofactor sign.

Using a Sparse Row in a Four-by-Four Matrix

This example separates two sign decisions. The outer factor \(-1\) is the cofactor sign for position \((2,3)\). Inside the minor, the first-row expansion has signs \(+,-,+\). Applying the outer sign only after the minor is calculated is a useful way to avoid mixing the two steps.

Using a Sparse Column in a Four-by-Four Matrix

How to Check the Work and Catch Sign Errors

A check is most useful when it tests steps that could have gone wrong. Expanding the same determinant along a different row or column changes which minors are used and which cofactor signs apply, so agreement provides a strong arithmetic check. It does not replace a careful calculation: both expansions could share an error if the same minor is copied incorrectly. For each minor, verify its entries by identifying the deleted row and column, and show the \(2\times2\) products explicitly.

Common mistake: forgetting the checkerboard sign. In the sparse-column example, the entry in position \((4,1)\) is \(3\), and its minor is \(-19\). A tempting but wrong calculation is to use \(+3(-19)\), as though every term in the expansion down a column had a positive sign. That gives \(-28-57=-85\), not the determinant. The correct position sign is \((-1)^{4+1}=-1\), so the term is \(-3(-19)=57\), and the determinant is \(-28+57=29\). The indices are the row and column numbers in the original matrix; do not renumber them after deleting a row or column.

A second common problem is deleting only one of the required indices. For \(M_{ij}\), delete row \(i\) and column \(j\), then keep the remaining entries in their original relative order. In the row-2 expansion of \(\mathbf{A}\), deleting row 2 and column 3 leaves rows 1, 3, and 4 and columns 1, 2, and 4. The resulting minor is therefore \(\begin{bmatrix}1&2&1\\2&-1&4\\1&0&-1\end{bmatrix}\), not a matrix formed by dropping the third row or by retaining column 3.

When choosing a direction, count the nonzero entries in each row and column and prefer one with fewer entries if it makes the arithmetic manageable. A first-row expansion is still useful when the problem asks for it or when its minors are simple. The goal is not to apply a special rule to every matrix, but to choose a valid expansion and keep its bookkeeping exact. The same cofactor formula works for any selected row or column.