Minors and Cofactors

4.4 Minors and Cofactors

The expansion of a determinant can be written compactly using minors and cofactors, which express it as a sum of products of elements with specially defined coefficients. These ideas form the foundation for the adjoint and inverse of a matrix.

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Minor of an Element

For any element $a_{ij}$ in a determinant, its minor is the determinant that remains after deleting the $i$-th row and the $j$-th column — the row and column in which the element lies.

Important: The minor of an element in a determinant of order $n$ (where $n \geq 2$) is itself a determinant of order $n-1$.

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Cofactor of an Element

The cofactor of an element $a_{ij}$ is simply its minor multiplied by a sign factor that depends on the position $(i, j)$.

The sign $(-1)^{i+j}$ gives a checkerboard pattern of signs:

• If $i+j$ is even, the cofactor equals the minor.
• If $i+j$ is odd, the cofactor equals the negative of the minor.

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Expansion of a Determinant Using Cofactors

Expansion Along the First Row

For the $3 \times 3$ determinant $\Delta$ above, expanding along row 1 gives:

Using cofactor notation, this becomes:

The determinant can also be expanded along any other row or column, and all six expansions give the same value:

• Along row 2: $\Delta = a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23}$
• Along row 3: $\Delta = a_{31}A_{31} + a_{32}A_{32} + a_{33}A_{33}$
• Along column 1: $\Delta = a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31}$
• Along column 2: $\Delta = a_{12}A_{12} + a_{22}A_{22} + a_{32}A_{32}$
• Along column 3: $\Delta = a_{13}A_{13} + a_{23}A_{23} + a_{33}A_{33}$

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A Critical Property: Multiplying Elements with Cofactors of a Different Row/Column

This is not an alternative way to compute the determinant — it is a separate result used later in finding the inverse of a matrix.

Proof of the Property

Consider the sum:

Here, we are taking elements from row 1 ($a_{11}, a_{12}, a_{13}$) and multiplying them by cofactors of row 2 ($A_{21}, A_{22}, A_{23}$).

Write out the cofactors explicitly:

Now substitute into $\Delta'$: …