Computing Determinants Using Row Operations
Properties of Determinants
Let's summarize the properties of determinants that we have learned so far: Let be n x n matrices.
- A determinant of a matrix is "linear" in any of the column vectors of the matrix i.e and .
- Suppose an n x n matrix consists of a column of zeros, then its determinant is zero because by (2), .
- Suppose an n x n matrix consists of two identical columns, then its determinant is zero because by (3), , which implies that .
- Let be an invertible matrix. Then . By (4) and (1), we have . Hence and .
Computing determinants
We have already learned that row operations can be used for computing the inverse of a matrix. In fact, the same procedure can be used for computing the determinant of the matrix as well. First of all, we let be an n x n elementary matrix. It is easy to see that
Given an invertible matrix , we use a sequence of row operations to transform it into an identity matrix. Let be the corresponding elementary matrices. Then we have and we have
Recall that we used row operations to find the inverse of . See here for details.
In the process, we used 1 row interchange, 6 row replacements and 2 row scalings ( and were used as factors). Then we have
Remarks:
- Since for any square matrix , similar results are also valid for "column operations".
- In fact, to compute the determinant of a matrix , it is enough to transform into an upper triangular matrix . Then .
- Here is the online tool for calculating determinants in "Linear Algebra Toolkit" developed by P. Bogacki.
Question
If is a non-invertible matrix, what can you say about ?