THE DETERMINANT

1. THE DETERMINANT 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 …

2. 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.

3. For any square matrix we denote by the square submatrix obtained from by deleting the 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:

4. For a 1x1 matrix we define For a 2x2 matrix we define For an arbitrary matrix we define recur-sively

5. or in MathSpeak (more concise and precise) Let’s compute a determinant. Let Note that Therefore

6. 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 computed correctly before. A standard notation for is And get

7. In the formula the number The right hand side of formula is called the By analogy we can think of an or even

8. One of the many marvelous properties of the function we called “determinant” is the theorem Theorem. Let be an matrix. For every Note that the theorem says that one can compute the determinant of A by using “expansion by cofactor”acrossany row or downany column.

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

10. Theoretically one of the important consequences of the theorem is the Corollary. 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.

11. 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.) • Exchange two rows. The determinant changes sign. • Multiply a row by a scalar c . The determinant gets multiplied by c . • 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

12. 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

13. Note that the previous result splits the set of square matrices (already layered by dimension) into three subsets: • Those such that • Those such that For later purposes we split the set of invertible matrices into Those with Those with

14. We state three useful properties of the determinant, two of which have easy proof by induction, and we leave the third unproven. • (see p. 172 of the textbook) 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.