We summarize the properties of the determinant that we already proved, and prove that a matrix is singular if and only if its determinant is zero, the determinant of a product is the product of the determinants, and the determinant of the transpose is equal to the determinant of the matrix.
We first defined the determinant of a matrix using cofactor expansion along the first row. (Definition def:toprowexpansion of DET-0010) We then introduced an alternative definition in terms of cofactor expansion along the first column and proved that the two definitions are equivalent. (Definition def:firstcolexpansion1 of DET-0020) We also proved the following simple but useful results:
We also found that elementary row operations affect the determinant as follows:
(Theorem th:elemrowopsanddet of DET-0030)
In this module we will prove the following important results:
To get us started, we need the following lemma.
Case 1. Suppose is obtained from by interchanging two rows, then so
Case 2. Suppose is obtained from by multiplying one of the rows of by a non-zero constant , then so
Case 3. Suppose is obtained from by adding a scalar multiple of one row to another row, then so
Recall that we first introduced determinants in the context of invertibility of matrices. Specifically, we found that is invertible if and only if . (A logically equivalent statement is: is singular if and only if .) We are now in the position to prove this result for all square matrices.
Conversely, suppose , then But then , so is singular.
Now suppose that is not invertible. Then is also not invertible. So, and . Thus .
The following theorem is a nice consequence of Theorem th:detofproduct. We leave the proof to the reader. (Practice Problem prob:proofdetofinverse)
In Practice Problem prob:dettranspose of DET-0020 you proved that . Your proof most likely relied on the fact that cofactor expansion along the first row produces the same result as cofactor expansion along the first column. (Theorem th:rowcolexpequivalence). We will now take another look at this result and prove it without the assumption that the two cofactor expansions produce the same outcome.
Now consider Clearly , but what is really important is that is also an elementary matrix. While was obtained from the identity by adding 4 times the third row to the first row, was obtained from the identity by adding 4 times the first row to the third row. By Theorem th:elemrowopsanddetitem:addmultotherrowdet of DET-0030, we know that .
So, for all three matrices we have .
We can generalize our observations in Exploration init:detoftranspose as follows:
We will now combine the three parts of our generalization into a lemma.
Case 1. Suppose is obtained from the identity by switching rows and . Then is the same as the identity matrix, except that and entries of are zero, and has a 1 in and spots. When the transpose is taken, 1’s and 0’s along the diagonal stay in place, while the -entry becomes the -entry and the -entry becomes the -entry. This shows that , and .
Case 2. Suppose is obtained from the identity by multiplying one of the rows by a non-zero constant. Observe that is a diagonal matrix, so and .
Case 3. Suppose is obtained from the identity by adding a times row to row . Then is the same as the identity matrix, except that the -entry of is . If we take the transpose of , then will become the -entry of the transpose. This means that can be obtained from the identity by adding times row to row . This means that .
We are now ready to prove the main result of this section.
Suppose is nonsingular. Then is can be written as a product of elementary matrices (Theorem th:elemmatrices of MAT-0060) By Theorem th:transposeproperties of MAT-0025 By Lemma lemma:detofelementarymat, for .