1. MATH 1046 Determinants (Section 4.2)
Alex Karassev

2. Determinant in the 2x2 case
Recall:
A is invertible if and only if det A ≠ 0

3. Determinant in the 2x2 case
Each product contains exactly one element from every column and every row

4. Determinants in the 3x3 case
Products of elements of the matrix
Every product contains exactly one element from each row and each column
Signs of products must alternate according to a special rule

5. Entries from the first row:
a11
a12
a13
a11a22a33
a12a23a31
a13a22a31
a11
a21
a11
a11a23a32
a12a21a33
a13a21a32

6. The first indices in all products are always 1,2,3 while the second indices form all possible rearrangements of 1,2,3
a11a22a33
a12a23a31
a13a22a31
a11
a21
a11
a11a23a32
a12a21a33
a13a21a32

7. How do we choose signs? a11a22a33 a12a23a31 a13a22a31 a11 a21 a11

8. Permutations
A permutation i1 i2 … in of 1,2,…,n is a rearrangement of 1,2,…,n
There are 12 … (n-1) n = n! permutations
We can assign a signature to each permutation as follows:
sgn(i1 i2 … in ) = (-1)k, where k is the number of two-element swaps (called transpositions) required to get i1 i2 … in from 1 2 … n

9. Remarks Choice of k is not unique For example:
can be obtained from in many different ways:

10. Remarks Choice of k is not unique For example:
can be obtained from in many different ways:

11. Remarks Choice of k is not unique For example:
can be obtained from in many different ways:

12. Remarks Choice of k is not unique For example:
can be obtained from in many different ways: →

13. Remarks Choice of k is not unique For example:
can be obtained from in many different ways: →

14. Remarks Choice of k is not unique For example:
can be obtained from in many different ways:
→ → (k=2)

15. Remarks Choice of k is not unique For example:
can be obtained from in many different ways:
→ → (k=2)
→ → → → (k=4)

16. Remarks Choice of k is not unique For example:
can be obtained from in many different ways:
→ → (k=2)
→ → → → (k=4)
However, in both cases the parity of k is the same, so sgn ( ) = (-1)2 = (-1)4 = 1
It can be shown in general that signature is well-defined (i.e. independent on k)

17. Definition of Determinant

18. Determinant – 3x3 case

19. Example

20. Example

21. Cofactor expansion – 3x3 case

22. Cofactor expansion – 3x3 case

23. Minors and cofactors
Let Aij be the (n-1)x(n-1) matrix obtained from A by deleting i-th row and j-th column
The (i,j)-minor of A is Mij = det Aij
The (i,j)-cofactor of A is Cij = (-1)i+j Mij

24. Minors and cofactors

25. Laplace cofactor expansion theorem
For all i,j = 1,2,…n we have
Cofactor expansion along the i-th row:
Cofactor expansion along the j-th column:
(see text for the proof)

26. Signs
(-1)#row + #column

27. Example

28. Example
Using cofactor expansion along the 3rd row:

29. Exercise
Make sure that cofactor expansion along the 2nd column gives the same answer

30. Theorem
Determinants of lower or upper triangular matrices are equal to the products of the diagonal elements
Consequently, the determinant of a diagonal matrix is the product of its diagonal elements

31. 4x4 case

32. Proof
Follows from cofactor expansions along the first row or the first column and induction on the size of a matrix

33. Determinants and the elementary row operations
Let B be obtained from A using one of the following operations:
Swap two rows: Ri  Rj
Multiply a row by a nonzero number: cRi
Add a multiple of one row to another row: Rj → Rj + cRi

34. Determinants and the elementary row operations
Swap two rows: Ri  Rj
det B = - det A
Multiply a row by a nonzero number: cRi
det B = c det A
Add a multiple of one row to another row: Rj → Rj + cRi
det B = det A

35. Corollary
Suppose that A ~ B. Then det A = 0 if and only if det B = 0

36. Proof – swapping two rows
B is obtained from A by swapping rows 1 and 2

37. Proof - multiplying a row by a nonzero scalar
B is obtained from A by cR1

38. Proof – adding a multiple of one row to another row
B is obtained from A by R1 → R1 + cR2

39. Corollary
A is invertible if and only if det A ≠ 0 (indeed, A is invertible if and only if the r.r.e.f. of A is I)
Example: the following matrix is invertible, since its determinant is 20 ≠ 0

40. Corollary
If A has a row (or column) consisting entirely of zeros or two coinciding rows (or columns) then det A = 0

41. Example
Since the following matrix is not invertible, its determinant is zero:

42. Determinants of elementary matrices
Recall:
Ri  Rj: Eij
cRi: Ei(c)
Rj → Rj + cRi : Eij(c)

43. Determinants of elementary matrices
Since the elementary matrices are obtained from I using elementary row operations and det I = 1, we have the following formulas
det(Eij)= -1
det(Ei(c))= c
det(Eij(c))= 1

44. Lemma
Let A and B be two n x n matrices. Then A or B is not invertible if and only if AB is not invertible

45. Proof Consider the following sequence of equivalent statements:
AB is not invertible
null(AB) ≠ 0
There exists x ≠ 0 in Rn such that (AB)x = 0
So A(Bx) = 0
If y = Bx = 0 then null(B) ≠ 0 so B is not invertible
Otherwise, y ≠ 0 and Ay = 0 so null(A) ≠ 0, and hence A is not invertible

46. Corollary
det (AB) = 0  det A =0 or det B =0

47. Determinants and matrix multiplication
Lemma If E is an elementary matrix and A is any n x n matrix then det (EA) = det(E)det(A)
Proof: follows from the effect of elementary row operations on determinants and from the formulas for the determinants of elementary matrices

48. Determinants of products
For n x n matrices A and B det (AB)= det(A)∙det (B)

49. Proof
If det A =0 it is the consequence of the previous corollary, so suppose A is invertible
Then A is a product of elementary matrices, A = A1 A2 … As, so we have:
det (AB) = det (A1 A2 … As B) = (by Lemma) = det (A1) det (A2 … As B) = (applying Lemma again) = det (A1) det (A2) det (A3… AsB) = (and so on) = det (A1) det (A2) … det (As-2) det (As-1) det (As) det B = (applying lemma in the other direction) det (A1)det(A2)…det(As-2) det(As-1As) det B = (and so on) =det(A1 A2 …As) det B = det(A)det(B)

50. Corollary If A is invertible then det (A-1) = 1/det(A) Proof
1 = det I = det(AA-1) = det(A) det(A-1)
Therefore det(A-1) = 1 / det (A)

51. Example Let I be the 2 x 2 identity matrix
Then det (I+2I) = det (3I) = 33 =9, while det(I)=1 and det(2I) = 22 =4
So in general det (A+B) ≠ det(A) + det (B)

52. Determinants of scalar multiples
For an n x n matrix A, det (cA) = cn det (A)
Proof cA = (cI)A and therefore det(cA) = det((cI)A) = det(cI) det(A) = cn det(A)

53. Determinant of transpose
det (AT) = det (A)

54. Proof Note that (AT)ij = (Aji)T
Using cofactor expansion along the first row of B=AT and induction on the size of the matrix we get:

55. Determinants and matrix operations - summary
det AB = det A det B = det BA
det (cA) = cn det A
det A-1 = 1 / det A
det AT = det A

56. Exercise
Find the following determinants:

57. Geometry of determinants
Let A be a 2 x 2 matrix
|det(A)| is the area of the parallelogram spanned by the columns of A
Similar interpretation exists in higher dimensions

58. Geometry of determinants
|det A|
Exercise: prove it! b c a

59. Example
Find the area of the triangle with vertices A(1,1), B(2,2),C(0,5)
Solution:
Since the area of triangle ABC is half the are of the parallelogram spanned by vectors AB and AC, we have:

60. Determinants and systems of linear equations
To solve Ax = b we can use Cramer’s rule:
Let Ai(b) be the matrix obtained by replacing the i-th column of A with b, i=1,2,…,n
Then the solution of the system is given by xi = det (Ai(b)) / det A, i = 1,2,…,n

61. Example
Solve the following system using Cramer’s rule:

62. Solution