# 2. Large Determinants

by M. Bourne

Evaluating large determinants can be tedious and we will use computers wherever possible (see box at right). But if you **have** to do large determinants on paper, here's how.

## Expanding 4×4 Determinants

### Computers and Determinants

Now that we have powerful tools like MathCad, Scientific Notebook, Mathematica, Matlab, Maple, etc, we should concentrate on understanding the uses of mathematics and not so much on its mechanics.

Students spend hours multiplying out large determinants and a lot of the time is spent figuring out where errors were made. And for what? That time is better spent using a computer (or calculator) to solve it directly for us and using the remaining time learning how to **apply** it.

Mostly, we will use Computer Algebra Systems to find large determinants.

We will use the same approach that we saw in the last section, where we expanded a 3×3 determinant.

Going down the first column, we find the cofactors of each element and then multiply each element by its cofactor.

`|(a,b,c,d),(e,f,g,h),(i,j,k,l),(m,n,o,p)|`

`=a|(f,g,h),(j,k,l),(n,o,p)| -e|(b,c,d),(j,k,l),(n,o,p)|` `+i|(b,c,d),(f,g,h),(n,o,p)|` `-m|(b,c,d),(f,g,h),(j,k,l)|`

Notice the pattern of

First term: positive

Second term: negative

Third term: positive

Fourth term: negative

To get the final answer, we expand out these 3×3 determinants, multiply then simplify.

Continues below ⇩

### Example

Expand the 4×4 determinant:

`|(7,4,2,0),(6,3,-1,2),(4,6,2,5),(8,2,-7,1)|`

As mentioned above, we will normally use a computer to find the value of large determinants.

### Search IntMath, blog and Forum

### Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Go to: Online algebra solver

### Algebra Lessons on DVD

Math videos by MathTutorDVD.com

Easy to understand algebra lessons on DVD. See samples before you commit.

More info: Algebra videos

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!