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)|`

Answer

The first step is to find the cofactors of each of the elements in the first column. We then multiply by the elements of the first row and assign plus and minus in the order:

plus, minus, plus, minus

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

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

Now, we expand out each of those 3 × 3 cofactors using the method that we saw before:

`| (3,-1,2), (6,2,5), (2,-7,1) |=15`

`| (4,2,0), (6,2,5), (2,-7,1) |=156`

`| (4,2,0), (3,-1,2), (2,-7,1) |=54`

` | (4,2,0), (3,-1,2), (6,2,5) |=-42`

Now, putting it all together, we have:

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

`=7xx15 - 6xx156` ` + 4xx54` ` -8xx-42`

`=-279`

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