1. Section 3.1 The Determinant of a Matrix

2. Determinants are computed only on square matrices. Notation: det(A) or |A| For 1 x 1 matrices: det( [k] ) = k

3. Determinants are computed only on square matrices. Notation: det(A) or |A| For 1 x 1 matrices: det( [k] ) = k For 2 x 2 matrices: For larger matrices, we define a determinant in terms of cofactors.

4. Def. If A is a square matrix, then the ij minor, denoted M ij, is the determinant of the matrix obtained by deleting the i th row and the j th column of A. The ij cofactor, denoted C ij, is given by C ij = (-1) i+j M ij

5. Ex. Find C 23 and C 13 for the matrix

6. Computing determinants by the cofactor expansion. The determinant of an n x n matrix A can be computed by expanding along the i th row: The determinant of an n x n matrix A can be computed by expanding along the j th column:

7. Ex. Compute the determinant of

9. Triangular matrices: If A is a triangular matrix then det(A) = a 11 a 22 a 33 · · · a nn

10. Triangular matrices: A = det(A) =

11. Section 3.2 Evaluation of a Determinant Using Elementary Operations

12. We computed determinants by the “cofactor expansion method” in the previous section. We shall introduce a new method which involves placing a given matrix into triangular form via elementary row operations. Why even bother with a second method for computing determinants if we already have one that works?

13. There are some problems in math that are theoretically simple but practically impossible. Think, for example, of a determinant of a 50 x 50 matrix. When computed by expanding by cofactors, this involves : 50 different 49 x 49 determinants. Each one of these 49 x 49 determinants requires 49 different 48 x 48 determinants. Each one of these 48 x 48 determinants requires 48 different 47 x 47 determinants. Each one of these.... We end up with a total of 50∙49∙48∙47∙ ∙ ∙6∙5∙4∙3 different 2 x 2 determinants (this is about 10 64 2 x 2 determinants that must be calculated). Even if a computer could calculate one million 2 x 2 determinants per second, it would take about 10 58 seconds (about 10 50 years) to finish calculating our 50 x 50 determinant. (According to the big bang theory, the universe is only about 10 10 years old.)

14. Order nCofactor Expansion Row Reduction Additions Multiplications Additions Multiplications 3 5 9 5 10 5119205 30 45 10 3,628,799 6,235,300285 339

15. Suppose that B is the triangular matrix obtained from A through row operations. We need to exploit the relationship between det(B) and det(A). To do so, we must first see how each row operation affects the value of a determinant.

16. Theorem: Suppose that A* was obtained from A through a single elementary row operation. i. If that operation was R i ↔ R j then we have: det(A*) = –det(A). ii. If that operation was R i + cR j → R i then we have: det(A*) = det(A). iii. If that operation was cR i → R i then we have: det(A*) = c det(A).

17. Ex. Verify iii. above is true on the following matrices:

18. Ex. Suppose. Compute the determinants of the following matrices. (a)

19. Ex. Suppose. Compute the determinants of the following matrices. (b)

20. Ex. Suppose. Compute the determinants of the following matrices. (c)

21. Ex. Use elementary row operations to compute the determinant of

22. If det(A) = 0, what do we know about the triangular matrix obtained by applying row operations on A?

23. If det(A) ≠ 0, then there is only one solution to a system represented by AX = B.

24. Section 3.3 Properties of Determinants

25. Let and. Then AB is. Compute: det(A) det(B) det(AB)

26. Theorem: Suppose A and B are n x n matrices and c is a scalar. 1. det(AB) = _________________ 2. det(cA) = _________________ 3. det(A T ) = _________________ det(A) det(B) c n det(A) det(A)

27. Theorem: det(A -1 ) = ___________

28. Theorem: If A is invertible then det(A) ≠ 0.

29. Theorem: Let A be an nxn matrix. The following are equivalent: 1. A is invertible. 2. AX = B has a unique solution. 3. AX = O has only the trivial solution. 4. rref(A) = I. 5. det(A) ≠ 0.

30. Section 3.5 Applications of Determinants

31. Cramer’s Rule:

32. Consider the following system of linear equations represented by the matrix equation AX = B: a 11 x 1 + a 12 x 2 + · · · + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + · · · + a 2n x n = b 2 a 31 x 1 + a 32 x 2 + · · · + a 3n x n = b 3 : :: a n1 x 1 + a n2 x 2 + · · · + a nn x n = b n Now, think of A in terms of its column vectors: A = [ a 1 a 2 a 3 · · · a n ] Define A 1 = [ b a 2 a 3 · · · a n ] A 2 = [ a 1 b a 3 · · · a n ] A 3 = [ a 1 a 2 b · · · a n ] : :: A n = [ a 1 a 2 a 3 · · · b ]

33. If det(A) ≠ 0 then there is a unique solution to AX = B which can be computed by:

34. Ex. Use Cramer’s rule to solve the following: x – y + 3z = 2 2x + y – z = 5 –x + y – 4z = –4