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 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 that 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.
The following theorem generalizes our observations.
The proof of this theorem is relegated to Tedious Proofs Concerning Determinants. For a sketch of the proof, you can watch this video:The following lemma is a useful consequence of parts item:rowswapanddet and item:rowconstantmultanddet of Theorem th:elemrowopsanddet.
We will prove Part item:det0lemma1. Parts item:det0lemma3 and item:det0lemma2 are left as exercises.
Because , we have the following counterpart of Theorem th:elemrowopsanddet for columns.
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). This gives us Therefore .
The following table summarizes the effect of each elementary row operation on the determinant.
Since the determinant of the row-echelon form of in (eq:refstep5) is , we have Therefore You should verify this result by direct computation using cofactors.