ch21

Determinants

Goals: To show how the determinant arises in the calculation of an nxn system of equations.
To calculate the determinant of an nxn matrix.

I. The Pattern

Historically, the concept of the determinant emerged from the algebraic patterns found by solving generalized n x n linear systems of equations. By finding general solutions to n x n systems of linear equations a definite algebraic pattern was discovered. To find the is pattern we will consider 1 x 1, 2 x 2 and 3 x 3 systems of equations.

1 x 1 Systems

Solve the system:

The coefficient matrix

We find:

Define: Det(A) = a, where Det(A) denotes the determinant of the 1x1 matrix A.

2 x 2 Systems

Solve the system:

The coefficient matrix

We solve the system

by elimination.

Step 1

Identify the row operation.

Step 2

Step 3

From Step 3:

If we continue we find:

Define: Det(A) =

for the 2 x 2 matrix A.

Notice that the determinant of A appears as the denominator of the solution for both the x and the y variables.

Akamai: If

a unique solution to the system exists.

3 x 3 Systems

Given a generalized 3 x 3 system:

Eq 1

The coefficient matrix

Eq 2

Eq 3

To solve this system, let

We solve this system using the Gaussian algorithm.

With

as the starting pivot we compute:

We now use the 2, 2 position for our pivot and (after an easy bit of algebraic manipulation) we compute:

(The 1 was put into the 3,3 position to make the matrix readable?)

That is,

If we solve for x and y, we notice that the denominators are identical. Thus the system has a solution if

then compare the denominator of the algebraic

expression for z above with

Define : Det(A) =

They are all identical!

Again, the Determinant of the coefficient matrix A appears as the denominator of the generalized solution.

Conjecture: ?

II. The Determinant of a Matrix

The determinant is defined:

For the 1 x 1 System,

the coefficient matrix is

The determinant of A

For the 2 x 2 System,

the coefficient matrix is

The determinant of A is:

For the 3 x 3 System,

the coefficient matrix is

The determinant of A is:

Informal Definition: Given the generalized n x n system of equations with the associated coefficient matrix

, the term "determinant of A" is the "denominator" formed when solving the associated system for each of the variables.

Akamai: An nxn linear system of equations has a unique solution iff the determinant of the
coefficient matrix is not equal to zero.

Once the determinant is defined, notice that solutions for the 2 x 2 system

could be written as:

Likewise solutions for the 3 x 3 System could be written as:

This rather remarkable result, that solutions to Ax = b could be written as formulae using determinants,
is called Cramer's Rule.

"In the eighteenth century, and even somewhat earlier, determinants of square arrays of numbers were calculated and used, often in the solution of systems of linear equations" [1]

Since the algebra of finding determinants by solving the associated system of equations is rather tedious, we would like to find an easier method to evaluate the determinants used in solving any given n x n system.

III. Calculation of |A|

When the concept of the determinant was formulated in 18th century Europe, several methods for calculating the determinant of a matrix were developed. We will look at two methods.

The first method uses permutations, while the second method was developed by the French mathematician Cauchy using what he called "minors and cofactors".

The Method of Permutations Using Leibniz's Formula

If you look closely at the subscripts of each of the terms of the determinants as they appear in the denominator of the general solution, you may notice that the indices of the subscripts are permutations.

The permutations of 1 and 2 are sometimes written as:

and

. These permutations are associated with the products:

and

. Note how these products correspond to the determinant

of the 2 x 2 matrix. Furthermore, if we consider the six

permutations of 1, 2 and 3, namely:

and

a pattern begins to emerge. Compare these permutations with the products found in the determinant of the 3 x 3 matrix.

Two facts:

1) We can use permutations to write out formulas for the determinant of an n x n matrix simply by writing out the permutations of the n numbers, and adding or subtracting the products depending on whether the permutations are even or odd. (More on that in a moment.) But is this really practical? No! Since the determinant of an n x n matrix would require n! permutations. For example, to evaluate a 10 x 10 matrix using permutations would require

permutations

2) Although such a formula tells us how to evaluate an n x n determinant, it is almost useless for hand calculation.

Example: Write out the formula for the determinant of a 4 x 4 matrix A.

There are 4! = 24 terms in the expression that correspond to the permutations of the numbers 1, 2, 3 and 4. Let's list a few of them with the corresponding terms:

etc.

These are five of the twenty-four terms of the determinant of a 4 x 4 matrix. The sign of the permutation depends on whether the permutation is even or odd, and may be found using a crossing diagram.
Example,

No crossing

1 crossing

3 crossings

2 crossings

Even

Odd

Even

sign +

sign -

sign +

Definition: The Complete or Permutation Expansion of A

For a general n x n matrix, the determinant was defined by Gottfried Leibniz with what is now known as the
Leibniz formula:

The sum is computed over all permutations s of the numbers {1, 2, ... , n} and sgn(s) denotes the sign of the permutation, sgn(s) = + 1 if s is an even permutation and sgn(s) = -1 if it is odd.

This definition is useful for proving the properties of determinants, but is not used for actual calculation.

This represents the awesome power of the algebra technique in revealing the underlying patterns found in mathematics. To use the complete expansion of permutations on a 10 x 10 system to find the determinant, oh mathfolk, be like counting one third of the cattle on the sun.

How do we calculate a determinant?

Laplace's Cofactor Calculation of |A|

If you learned how to calculate the determinant, it was probably the method of using "minors and cofactors". In practice this is the generally preferred method.

Definition: The Cofactors and Minors of A
Let A = (a_ij) be an nxn matrix. Let M_ij be the (n-1)x(n-1) matrix
obtained from A by deleting the row and column containing a_ij. The determinant of M_ij is called the minor of a_ij. The cofactor C_ij of a_ij is defined as

Calculation of Minors using a Mathcad subroutine

For example, let

and

then

The minor is composed of the entries of A
that are not in row 3 or column 1.

the minor of

and the Cofactor

is calculated to be

where the absolute bars denote the determinant operator Det(M).

For the case of the 3 x 3 System we found

in terms of minors

this may be written:

In terms of cofactors

we may write:

Eq 6

Problem: Use Eq 6 to calculate the determinant of

1) Calculate the

's.

2) Plug these numbers into

Eq 6.

Check

Putting these ideas together we may define a determinant recursively as follows.

The Determinant of A: The determinant of an n x n matrix A = (a_ij) denoted Det(A) , is a scalar number that is associated with the matrix A that is defined recursively as follows:

where

Akamai: This definition allows you to expand a determinant about any row or column.

While we will not prove that this method of calculation is equivalent to the permutation expansion, we will check the 3 x 3 case.

Show that the cofactor expansion for a 3 x 3 matrix A is equivalent to the determinant formula

Let

. Expand about any row or column. In this case we select the second row.

Write out

where the C_ij refer to the cofactors of A

after multiplying this out and rearranging, gives:

Once the determinant is defined, we begin a study of its properties. One important property that we will not explore immediately, is that the determinant defines a function from the set of n x n matrices to the set of real numbers.

Thinking like a mathematician:

What would the determinant be if the matrix A contained entries from the field of complex numbers, or a finite field like Z₇?

IV. A Few Theorems

Theorem: If A is an nxn Matrix with n >1, then Det(A) can be expressed as a cofactor expansion using any row or column of A.

Proof: ?

For a row expansion, the cofactor formula looks like:

The column expansion has the form:

Theorem: If A is an nxn Matrix, then Det(A^T) = Det(A).

(Proof by Induction)

Notice that the rows of A are the columns of A^T

If n = 1.

Assume true for n = k.

Theorem: If A is an nxn Triangular Matrix, then the determinant of A equals the product of the diagonal elements of A.

Illustrate with an example.

Theorem : Let A be an nxn Matrix.

i) If A has a row or column consisting entirely of zeros, then Det(A) = 0.

ii) If A has two identical rows or two identical columns, then Det(A) = 0.

Proof: Exercise.