Elementary Linear Algebra: For University of Lethbridge Math 1410

In the previous section we learned how to compute the determinant. In this section we learn some of the properties of the determinant, and this will allow us to compute determinants more easily. In the next section we will see one application of determinants.

We start with a theorem that gives us more freedom when computing determinants.

We alluded to this fact way back after Example 6.3.6. We had just learned what cofactor expansion was and we practised along the second row and down the third column. Later, we found the determinant of this matrix by computing the cofactor expansion along the first row. In all three cases, we got the number 0. This wasn’t a coincidence. The above theorem states that all three expansions were actually computing the determinant.

How does this help us? By giving us freedom to choose any row or column to use for the expansion, we can choose a row or column that looks “most appealing.” This usually means “it has lots of zeros.” We demonstrate this principle below.

Wow. That was a lot simpler than computing all that we did in Example 6.3.10. Of course, in that example, we didn’t really have any shortcuts that we could have employed. Our next example involves a \(5\times 5\) determinant. At first, this looks like trouble, until we realize that the matrix is triangular. As we’ll see, this makes our job much easier.

We see that the final determinant is the product of the diagonal entries. This works for any triangular matrix (and since diagonal matrices are triangular, it works for diagonal matrices as well). This is an important enough idea that we’ll put it into a box.

It is now again time to start thinking like a mathematician. Remember, mathematicians see something new and often ask “How does this relate to things I already know?” So now we ask, “If we change a matrix in some way, how is its determinant changed?”

The standard way that we change matrices is through elementary row operations. If we perform an elementary row operation on a matrix, how will the determinant of the new matrix compare to the determinant of the original matrix?

Let’s experiment first and then we’ll officially state what happens.

We’ve seen in the above example that there seems to be a relationship between the determinants of matrices “before and after” being changed by elementary row operations. Certainly, one example isn’t enough to base a theory on, and we have not proved anything yet. Regardless, the following theorem is true.

Let’s put this theorem to use in a couple of examples.

In the previous example, we may have been tempted to “rebuild” \(\tta\) using the elementary row operations and then computing the determinant. This can be done, but in general it is a bad idea; it takes too much work and it is too easy to make a mistake.

Let’s continue to think like mathematicians; mathematicians tend to remember “problems” they’ve encountered in the past, and when they learn something new, in the backs of their minds they try to apply their new knowledge to solve their old problem. (This is why mathematicians rarely smile: they are remembering their problems)

What “problem” did we recently uncover? We stated in the last chapter that even computers could not compute the determinant of large matrices with cofactor expansion. How then can we compute the determinant of large matrices?

We just learned two interesting and useful facts about matrix determinants. First, the determinant of a triangular matrix is easy to compute: just multiply the diagonal elements. Secondly, we know that given any square matrix, we can use elementary row operations to put the matrix in triangular form. We can then find the determinant of the new matrix (which is easy), and adjust that number by recalling what elementary operations we performed.

In practice, we don’t need to keep track of operations where we add multiples of one row to another; they simply do not affect the determinant. Also, in practice, these steps are carried out by a computer, and computers don’t care about leading 1s. Therefore, row scaling operations are rarely used. The only things to keep track of are row swaps, and even then all we care about are the number of row swaps. An odd number of row swaps means that the original determinant has the opposite sign of the triangular form matrix; an even number of row swaps means they have the same determinant.

For larger determinants, we can follow the same routine: create zeros in one column, expand along that column to get a smaller determinant, and repeat.

Let’s think some more like a mathematician. How does the determinant work with other matrix operations that we know? Specifically, how does the determinant interact with matrix addition, scalar multiplication, matrix multiplication, the transpose and the trace? We’ll again do an example to get an idea of what is going on, then give a theorem to state what is true.

We now state a few theorems that confirm our conjectures from the previous example.

To see that Theorem 6.4.11 is true, note that if \(A\) is not invertible, then the reduced row echelon form \(R\) of \(A\) must have a row of zeros. Performing a cofactor expansion along this row, we immediately see that \(\det(R)=0\text{.}\) Since \(R\) is obtained from \(A\) by a series of elementary row operations, we know from Theorem 6.4.6 that \(\det(A)\) is a multiple of \(\det(R)\text{,}\) and thus \(\det(A)=0\text{.}\)

It follows from Theorem 6.4.11 (using the logical principle known as the contrapositive) that if \(\det(A)\neq 0\text{,}\) we’re guaranteed that \(A\) is invertible.

At this point, we naturally should ask whether or not the converse to Theorem 6.4.11 is true as well: suppose we know \(\det(A)=0\text{.}\) Does that imply that \(A\) is not invertible? (Or equivalently, if we know \(A\) is invertible, does this imply that \(\det(A)\neq 0\text{?}\)) The answer is yes, but to see this, we first need a more general result.

Proving that \(\det(AB)=\det(A)\det(B)\) is most easily done using elementary matrices. (See Section 4.6.) Recall that multiplying a matrix on the left by an elementary matrix is the same as doing the corresponding row operation: if \(A\) is any \(3\times 3\) matrix, then \(EA\) can be obtained from \(A\) using the same row operation used to create \(E\text{.}\)

As an added bonus, we can rearrange the above equation to give us one more property of the determinant:

Combining Theorems Theorem 6.4.11 and Theorem 6.4.13 allows us to add on to our Invertible Matrix Theorem.

This new addition to the Invertible Matrix Theorem is very useful; we’ll refer back to it in Chapter 7 when we discuss eigenvalues.

In the next section we’ll see how the determinant can be used to solve systems of linear equations.