5.2 Using Gaussian Elimination

The determinant can also be found using EROs in a manner similar to Gaussian elimination. In order to use EROs to find the determinant of a matrix, you must know a few facts about the determinant:

1. Interchanging two rows changes the sign of the determinant.

2. Multiplying a row by a scalar multiplies the determinant by that scalar.

3. Replacing any row by the sum of that row and any other row does NOT change the determinant.

4. The determinant of a triangular matrix (upper or lower triangular) is the product of the diagonal elements.

You can demonstrate these facts to yourself using a 2 by 2 matrix. Just as using EROs does not change the solution to a system, EROs combined with these rules will allow us to find the determinant of the original matrix. The use of EROs results in a system that is equivalent to the original system, so if we apply these rules to the determinant as we change the system, we will find the determinant to the original coefficient matrix. Let's use these rules to find the determinant of a 2 by 2 matrix. As we work this problem, we will let D_i represent the determinant of the current matrix yielded by EROs

When we check this result using the formulas that we know, we get det(A) = 5(-2) - (-4)3 = 2. Actually, we can solve for det(A) at any of the steps, but we work until we have a diagonal matrix because the determinant of a diagonal matrix is easy to find since it is simply the product of the diagonal elements. Let's use this method to find the determinant of a 3 by 3 matrix.

We can verify this with either of the methods that we learned earlier to find the determinant of a matrix. Using only these small examples might lead you to think that it is slower to find the determinant using elementary row operations. However, it actually requires fewer steps for larger problems than does expansion by minors. Therefore, this method is used more often to find the determinant in computational mathematics.