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.
Let
Find by cofactor expansion along the second row.
Matrix is the same as the matrix
in Example ex:expansiontoprow of DET-0010. According to the Laplace Expansion Theorem we should
get the same value for the determinant as we did in Example ex:expansiontoprow regardless of which row
or column we expand along. The second row has the advantage over other
rows in that it contains a zero. This makes computing one of the cofactors
unnecessary. Following the checker board sign pattern along the second row we
get
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.
Find if
The fourth column contains the most zeros, so we will expand along that
column. The -entry is the only non-zero entry in the fourth column. So the
sign is given by . You can also consult the checker board pattern to confirm
this.
Next we will expand along the top row.
Try the next step on your own. We suggest that you expand the first matrix along
the last column and expand the second matrix along the first column.
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)