Determinants. Cramers Rule.

Determinants.

Let's introduce determinants.

A nth-order determinant is an expression that is always associated with a nxn matrix.

Note: this will be a square matrix!

Let us begin with 2^nd order Determinants.

A second order determinant D must be associated to a 2x2 matrix A:

If , then

Note: While the entries of the matrix A are written between rectangular brackets "[" and "]", the same entries for the determinant D are written between bars …

We say "entries for the determinant", since the determinant itself is just a scalar:

Often, the determinant associated with matrix A is written as in order to express this association!

Example:

Why?

Another Example: Use of determinants to solve systems of linear equations:

The system

has the solutions (if )

Now, this is nothing else but

with

This is called "Cramer's rule"

Let's verify this:

therefore

Conclusion: We can use determinants to solve systems of linear equations!

Use Cramer's rule in an example:

(The associated matrix is , then the equation reads )

In order to get D₁, we replace the first column of D with the values of b:

And, in order to get D₂, we replace the second column of D with the values of b:

And, according to Cramer's rule we have solutions:

Þ ,

So far second order determinants

3^rd order Determinants:

Let's move up to third order determinants. Associated to a third order determinant there will be a 3x3 matrix:

Again we write the entries of a between bars to point out this is a determinant, not a matrix! And how do we calculate the value of a 3^rd order determinant?

We select, for example, the first row of D.

For each entry of column 1, we find the "minor" determinant, this is a determinant consisting on all remaining entries excluding those from the first row and those from the particular column.

e.g.:

, etc.

We then multiply each entry of column 1 with it's minor and add them up, using alternating sign:

Since all three minors are 2x2 determinants, we can calculate their values with ease.

Now we are able to extend Cramer's rule to linear systems with 3 unknown

For the system

Cramer's rule gives us solutions

, , and

with

and, of course,

We now proceed and find a way to obtain the values of determinants of arbitrary order n.

This will lead us to a method to solve systems of n equations using Cramer's rule.

Determinants of Order n

for n=1 the associated matrix has one element only: We define the determinant of such a matrix to be the value of this entry. Far more interesting are determinants of higher order: .

We are actually able to formulate a precise recipe how to obtain the value of such a determinant of order n:

Let be

and

Then D can be expanded as

(where j can be any row subscript between 1 and n)

or as

(where k can be any column subscript between 1 and n)

where

with being the determinant of the submatrix of A obtained by eliminating row j and column k (the minor). is of the order n-1!

This looks more complicated as it is.

We simply pick a row or a column for which we want to "expand" the determinant.

This can be any row or column. It really doesn't matter which one!

Let's select, for example, row "j" (marked in red):

Now we have to find for each entry from row j the minor and multiply it with the entry and the factor .

The first entry would be , and the associated minor would consist on the entries marked in black (these are all entries except those from column 1 and from row j):

(Note that column 1 and row j are missing)

For another element in this row, let's say , we identify the entries of the minor as all those except the ones in row j and column k

and can be obtained as:

(Note: this time the entries from row j and column k are not included)

(expanding by row) or

(expanding by column)

All what's left to do is to determine the sign of each term through the factor

This is easy:

if the subscripts both are even or odd, the factor will be (+1)
if one subscript is even and the other is odd, the factor will be (-1)

· the factor leads to a checker board pattern of signs:

Let's try a simple example: Determinant of the order 3:

Example:

Expand by 1^st row:

Identify the minors

To :

and, consequently we obtain the determinant as

Now, let’s expand the same determinant by the 3^rd column – after all we claim the result is always the same:

Expand by 3^rd column:

The third column consists on the entries . We have to identify the corresponding minors:

To we already found :

And to

so we get

No matter by which row or column you expand this determinant, the result will be always the same. Such as "-12" in our example.

This behavior of determinants allows us to come up with some shortcuts:

Shortcuts:

· If possible, expand by a row/column that contains 0's

The entries of the row by which you expand are the factors you have to multiply the minors with. The more of these factors are zero, the less of these products will occur and the less minors you have to expand. Always look for rows/columns that contain a lot of zero’s!

· If a matrix is triangular, it's determinant is equal to the product of all diagonal elements (entries):

Although this is obvious we might add some explanation to it. For example, assume a determinant that is upper triangular …..e.g. the following 5^th order determinant:

Let us expand this determinant by the first column. This gives us:

Very similar, for a lower triangular determinant, we can do an expansion by the last column. (You might want to try this yourself). Again we will find that the determinant will be the product of all diagonal entries.

You can make another conclusion from this: If a determinant is triangular, and if one of the diagonal entries is equal zero, then the determinant will be zero!

Copyrights 1999, 2000 by Peter Dragovitsch and Ben A. Fusaro