# Elimination method to find the determinant

1. Mar 6, 2006

### kdinser

The instructions for the section say:

use the method of elimination to evaluate the determinants in problems 13-20.

They are all 3x3 or 4x4 matrices

I can't find an example of this in the book. Can someone outline the procedure for me? I've already solved these same problems using other methods described in the book and in class and had no problems. I just don't want to get caught with my pants down if he asks for this method to be used on the test. Thanks

2. Mar 6, 2006

### 0rthodontist

I think they just want you to use row reduction to simplify the matrices before expanding the determinant.

3. Mar 6, 2006

### xman

use coffactor expansion along any row or column, in particular, look for rows or column to expand about if it has a lot of zeros or ones. ex.
$$\mid \begin{array}{ccc} a & b & c \cr d & e & f \cr g & h & i \end{array} \mid = a \mid \begin{array}{cc} e & f \cr h & i \end{array} \mid- b \mid \begin{array}{cc} d & f \cr g & i \end{array} \mid +c \mid \begin{array}{cc} d & e \cr g & h \end{array} \mid$$
hope this helps, x

4. Mar 7, 2006

### HallsofIvy

Staff Emeritus
Cofactor expansion is not "method of elimination". Use row operations to reduce the matrix to a triangular matrix. Then the determinant is just the product of the numbers on the diagonal. One thing to be careful about: while "add or subtract a multiple of one row to another" will not change the value of the determinant, multiplying a row by a number will: you will have to divide the resulting matrix by that number to get the original matrix. Also swapping two rows will change the sign of the determinant.