We state and prove the Laplace Expansion Theorem for determinants.

### DET-0050: The Laplace Expansion Theorem

#### Introduction and Examples

We originally defined the determinant of a matrix in terms of cofactor expansion along the top row of the matrix. We later showed that cofactor expansion along the first column produces the same result. Surprisingly, it turns out that the value of the determinant can be computed by expanding along any row or column. This result is known as the Laplace Expansion Theorem. We begin by generalizing some definitions we first encountered in DET-0010.

Given an matrix define to be an matrix obtained from by deleting the row and the column of .

Define the *-cofactor* of by
Note that the sign of follows a checker board pattern.

This answer is the same as the answer we got using cofactor expansion along the first row in Example ex:expansiontoprow.

It is clear that having zeros as entries in the matrix significantly reduces the number of computations necessary to find the determinant. The following example demonstrates how Laplace Expansion Theorem allows us to use zeros to our advantage.

#### Proof of the Laplace Expansion Theorem

We started the topic of determinants by introducing two definitions of the determinant and proving that they produce the same result. However, if you examine our proofs carefully, you will find that none of the proofs rely on cofactor expansion along the first row. So, all of our results were derived based exclusively on cofactor expansion along the first column. Indeed, we did not have to present the first definition at all. Our only motivation for doing so was that both definitions are standard and widely used.

In keeping with our effort to avoid cofactor expansion along the first row in proofs, we will prove the Laplace Expansion Theorem using cofactor expansion along the first column. Then cofactor expansion along the first row will simply become a consequence of the Laplace Expansion Theorem, rendering the equivalence proof (Theorem th:rowcolexpequivalence) of DET-0020 redundant.

We begin by stating the following counterpart of Theorem th:elemrowopsanddet of DET-0030.

- Proof
- This is a direct consequence of the fact that . (Theorem th:detoftrans of DET-0040)

We are now ready to prove Theorem th:laplace1. For convenience we restate the result:

Let be an matrix. Then

and

- Proof of Laplace Expansion Theorem
- We will start by showing that cofactor
expansion along column produces the same result as cofactor expansion along the
first column. Observe that column can be shifted into the first column position by
consecutive row switches. Let be the matrix obtained from by performing the
necessary column switches. Then
To show that the determinant of can also be computed by cofactor expansion along any row follows from the fact that . (Theorem th:detoftrans of DET-0040)