We examine the effect of elementary row operations on the determinant and use row reduction algorithm to compute the determinant.
When we first introduced the determinant we motivated its definition for a matrix by the fact that the value of the determinant is zero if and only if the matrix is singular. We will soon be able to generalize this result to larger matrices, and will eventually establish a formula for the inverse of a nonsingular matrix in terms of determinants.
Recall that we can find the inverse of a matrix or establish that the inverse does not exist by using elementary row operations to carry the given matrix to its reduced row-echelon form. In order to start relating determinants to inverses we need to find out what elementary row operations do to the determinant of a matrix.
Recall that there are three elementary row operations:
Elementary row operations are used to carry a matrix to its reduced row-echelon form. In Practice Problem prob:elemrowopsreverse of SYS-0010 we established that elementary row operations are reversible. In other words if we know what elementary row operations carried to , we can undo each operation with another elementary row operation to carry back to . This will prove useful for computing the determinant. Computing the determinant of is easy. (Why?) If we know what elementary row operations carry back to , and what effect each of these operations has on the determinant of , we could find the determinant of .
Next, construct matrix by multiplying the last row of by . Find . It turns out multiplying the first or the second row of by yields exactly the same result as this.
Finally, construct matrix by adding twice row 3 to row 1. Find . This result is particularly surprising. Try a few more variations of this example to convince yourself that adding a multiple of one row to another row does not appear to affect the determinant.
We are now ready to state and prove a general result.
We proceed by induction on . The result is not applicable for . In Practice Problem prob:proofofrowswapanddet, you will be asked to verify that the result holds for matrices. Suppose that the result holds for matrices. We need to show that it holds for matrices. You may find the following diagram useful throughout the proof.
Observe that for we have: Because is obtained from by switching two rows of , our induction hypothesis give us:
For and we have:
We compute the determinant of by cofactor expansion along the first column.
If two non-adjacent rows are switched, then the switch can be carried out by performing an odd number of adjacent row interchanges (See Practice Problem prob:numberofrowswitches), so the result still holds.
Suppose is obtained from by multiplying the ’s row of by .
We compute the determinant of by cofactor expansion along the first column.
Before we tackle the proof of Part item:addmultotherrowdet of Theorem th:elemrowopsanddet we will need to prove the following two lemmas.
We will prove Part item:det0lemma1. Parts item:det0lemma3 and item:det0lemma2 are left as exercises.
You may find the following representations of , and helpful. Identical entries in , and are labeled .
Observe that For , the induction hypothesis gives us
We now compute the determinant of by cofactor expansion along the first column.
We are now ready to finish the proof of Theorem th:elemrowopsanddet
What we discovered about the effects of elementary row operations on the determinant will allow us to compute determinants without using the cumbersome process of cofactor expansion.
by Theorem th:elemrowopsanddetitem:addmultotherrowdet
by Theorem th:elemrowopsanddetitem:rowswapanddet
by Theorem th:elemrowopsanddetitem:rowconstantmultanddet
by Theorem th:elemrowopsanddetitem:addmultotherrowdet
Recall that (Lemma lemma:detofid of DET-0010). This gives us Therefore .
We stop at the row-echelon form of because its determinant is easy to compute by multiplying its diagonal entries (See Practice Problem prob:detoftrimat of DET-0020).
The following table summarizes the effect of each elementary row operation on the determinant.
Combining (eq:detofrefa) with the information from the table, we get Thus You should verify this result by direct computation using cofactors.