Consider the set of all square matrices. To actually compute the inverse of a square matrix A (actually to determine if the inverse exists and also compute it) is a rather long process involving the reduction of A to echelon form, through a sequence of elementary row operations, while applying the same sequence to the identity matrix.If one needs to know this is probably the easiest method (there are others), but often …

THE DETERMINANT

we just want to know that the inverse exists, without actually knowing the individual entries in it.This is one of the major purposes of computing determinants, we will be able to tell rather quickly and efficiently when the inverse exists.There is another major use of determinants that deals with areas, volumes (we will learn this) and integration by substitution in several variables (the jacobian, you will learn it in Calculus III)Let’s start by defining the determinant and learning how to compute it.We need some notations first.

For any square matrix we denote by the square submatrix obtained from by deletingthe Note that the dimension of is one less than the dimension of .We define the “determinant” as a function

defined in the following manner:The set of all square matrices can be “layered” by the dimension as shown in the figure:

For a 1x1 matrix we define For a 2x2 matrix

we define

For an arbitrary matrix we define recur-sively

or in MathSpeak (more concise and precise)

Let’s compute a determinant. Let

Note that

Therefore

The description given after Exercise 14 on p. 168 of the textbook is extremely useful. LEARN IT !I will use it here once to verify we computedcorrectly before. A standard notation foris

And get

In the formula

the number

The right hand side of formula is called the

By analogy we can think of an

or even

One of the many marvelous properties of the function we called “determinant” is the theoremTheorem. Let be an matrix. For every

Note that the theorem says that one can compute the determinant of A by using “expansion by cofactor” across any row or down any column.

The theorem, whose proof we omit, is very powerful, both computationally and theoretically. Here is a computation:

VOILÁ

Theoretically one of the important consequences of the theorem is theCorollary. If A is a square triangular matrix, then

Proof. Here are the four types of triangular matrices:

2 types, upper and lower 2 types, upper and lower apply the theorem as convenient.

The next property of the determinant function that we will study is how the determinant behaves under elementary row operations. Here is the behavior:(Stated in textbook as Theorem 3, p. 169.)1. Exchange two rows. The determinant changes

sign.2. Multiply a row by a scalar c . The determinant

gets multiplied by c .3. Add the multiple of one row to another row. The

determinant does not change.(Proof omitted here, in textbook on pp. 173-174)Theorem 3 has an extremely important consequence, namely

Theorem (4 in textbook, p. 171). A square matrix A is invertible if and only if Proof. Begin with the remark that any square matrix in echelon form is triangular. Now

Note that the previous result splits the set of square matrices (already layered by dimension) into three subsets:A. Those such that

B. Those such that

For later purposes we split the set of invertible matrices intoThose with

Those with

We state three useful properties of the determinant, two of which have easy proof by induction, and we leave the third unproven.

1. (see p. 172 of the textbook)2. 3. I have not seen statement 2. in the textbook. The first student who gives me a proof of 2.will get 3 extra points on the total homework grade.