Determinant

combining row operations with expanding by cofactors

The following describes the process of finding the determinant of a square matrix by the method of expanding by cofactors. What makes this different than any old 'expanding by cofactors' method is that here we make use of row operations on a determinant to make the resulting expansion simplier to evaluate.
Combining row operations with expanding by cofactors is not necessary in order to find a determinant, but row operations can sure make the follow-up steps easier. That is why we use them. While it is true that your calculator can find determinants really fast, it is difficult to have a appreciation for how something works if you have never worked on the insides of a problem. And that is part of what this class is about....appreciating the guts of algebra. So, on to the main point of this writing.

A determinant is a real number that is associated with a square matrix and so every square matrix has a determinant. The determinant is defined in terms of cofactors, which are also determinants. Using mathematical terminology states that we say that we 'expand about a row or column' until we have described the determinant in terms of the sums of 2 x 2 determinants.

You might recall that 2 x 2 determinants are really easy to evaluate. If you have forgotten how to do this, then check out the explanation in your text before you continue on here.

EXAMPLE 1:

Here is an example of a 4 x 4 matrix that has been expanded about column 3.

Notice that each entry in column 3 has been multipled by its cofactor and the results added.

The cofactor consists of the SIGN OF THE POSITION OF THE ENTRY and the determinant that is left over after you eliminate the row and the column of the entry.
The SIGN OF THE POSITION OF THE ENTRY can be determined by raising (-1) to the power equal to the sum of the row and the column of the entry.

Example: the entry in row 4, column 3 is 6.
The sign of the position of row 4, column 3 is (-1)⁽⁴⁺³⁾ = (-1) ⁷ = (-1)

Note: The entry in row 3, column 3 is 0 so when it is multipled by its cofactors the result will still be 0. It is unnecessary to include the 0 term in the expansion.

How can we make this process faster?

If we were to continue along this path then the next step would be to evaluate each of the 3x3 determinants by expanding about a row or column, generating a bunch of 2x2 determinants. But that gets to be a big job - one that we would just as soon avoid.
We use row operations on the determinant to generate a column or row with as many 0's in it as possible (you will end up with either1or 0 non-zero entries). The cofactor for the non-zero entry is the only that you will have to evaluate. We keep this process up until we have just one 2x2 determinant left.

Thus, any determinant can be row reduced down to a product of a real number and a 2x2 determinant. Here is an example.

Shorthand: To facilitate summarizing the steps, the following shorthand will be used:

3 R1 + R2 -----> R2 will mean:

multiply Row 1 by 3, add it to Row 2, and put the results in Row 2

EXAMPLE 2: Let's go back to the beginning of Example 1 and use row operations to generate 0's in column 3. I plan to keep entry (1,3) just as it is and use it to zero out the rest of column 3 [remember, entry (1,3) is the entry in row 1, column 3]. I chose this entry because it has a value of 1 and that will make the next steps easier. However, you can use any entry that you wish and you will get the same value for the determinant. After you read through this page you might want to come back and try to find the determinant using your own series of steps.

STEP 1:

(-1) R1 + R2 -----> R2

(- 6) R1 + R4 -----> R4

STEP 2:

Column 3 has only one non-zero entry so we expand about column 3, yielding the following:

Notice that the coefficient of the 3x3 determinant is just 1, so it is not necessary to write it at each following step. We are now going to keep entry (1,1) just like it is, and use it to zero out the rest of column 1.
FYI: I pick entry (1,1) here because it has a value of (- 1). Any of the entries could have been used but entries of 1 and -1 usually are easiest to work with.

STEP 3:

3 R1 + R2 -------> R2

-18 R1 + R3 -------> R3

STEP 4:
Expanding about column 1 gives us the following:

Therefore,

How does this process apply to larger matrices?

If the original problem had been to find the determinant of a 10x10 matrix then we would have performed row operations so that the first expansion would have yielded a single 9x9 determinant. After the second set of row operations we would have an 8x8 determinant and so on until we had only a single 2x2 determinant.
To see if you have mastered this process, why don't you take the original 4x4 determinant and evaluate it by using 2 or three different approaches. If you do it correctly then each determinant will be 72.

NOTE: Review your text on cofactors and finding determinants by expanding by cofactors. Pay particular attention to the fact that as you expand about a row or column you multiply each ENTRY by the SIGN OF THE POSITION OF THE ENTRY and then by the determinant as defined in your text.

© 1999 Jo Steig