This document contains notes from a lesson on determinants and inverses. It defines determinants as the sum of products of matrix elements with one element from each row and column, and lists several rules for how determinants change with row/column operations, including that the determinant is 0 if rows/columns are proportional or duplicated. It also states the determinant of a product of matrices is the product of the individual determinants.

1. Lesson 8
Determinants and Inverses (Section 13.5–6)
Math 20
October 5, 2007
Announcements
No class Monday 10/8, yes class Friday 10/12
Problem Set 3 is on the course web site. Due October 10
Sign up for conference times on course website
Prob. Sess.: Sundays 6–7 (SC 221), Tuesdays 1–2 (SC 116)
OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)

2. Review: Determinants of n × n matrices by patterns
Definition
Let A = (aij )n×n be a matrix. The determinant of A is a sum of
all products of n elements of the matrix, where each product takes
exactly one entry from each row and column.

3. Review: Determinants of n × n matrices by patterns
Definition
Let A = (aij )n×n be a matrix. The determinant of A is a sum of
all products of n elements of the matrix, where each product takes
exactly one entry from each row and column.
The sign of each product is given by (−1)σ , where σ is the number
of upwards lines used when all the entries in a pattern are
connected.

4. 4 × 4 sudoku patterns
+ − − + + −
− + + − − +
+ − − + + −
− + + − − +

6. Determinants of n × n matrices by cofactors
Definition
Let A = (aij )n×n be a matrix. The (i, j)-minor of A is the matrix
obtained from A by deleting the ith row and j column. This matrix
has dimensions (n − 1) × (n − 1).
The (i, j) cofactor of A is the determinant of the (i, j) minor
times (−1)i+j .

8. Fact
The determinant of A = (aij )n×n is the sum
a11 C11 + a12 C12 + · · · + a1n C1n

9. Example
2 −4 3
Compute the determinant: 3 1 2
1 4 −1
Expand along 1st row
Expand along 2nd row
Expand along 1st column

11. Fact
The determinant of A = (aij )n×n is the sum
a11 C11 + a12 C12 + · · · + a1n C1n
Fact
The determinant of A = (aij )n×n is the sum
a11 Ci1 + ai2 Ci2 + · · · + ain Cin
for any i.

12. Fact
The determinant of A = (aij )n×n is the sum
a11 C11 + a12 C12 + · · · + a1n C1n
Fact
The determinant of A = (aij )n×n is the sum
a11 Ci1 + ai2 Ci2 + · · · + ain Cin
for any i.
Fact
The determinant of A = (aij )n×n is the sum
a1j C1j + a2j C2j + · · · + anj Cnj
for any j.

13. Theorem (Rules for Determinants)
Let A be an n × n matrix.

14. Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| =

15. Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.

16. Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A | =

19. Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A | = |A|

21. Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A | = |A|
3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.

22. Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A | = |A|
3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.
4. If two rows or columns of A are interchanged,

24. Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A | = |A|
3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.
4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.

25. Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A | = |A|
3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.
4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| =

27. Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A | = |A|
3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.
4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.

28. Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.

29. Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then
|A| =

30. Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then
|A| = 0.

31. Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then
|A| = 0.
6. If a scalar multiple of one row (or column) of A is added to
another row (or column), then

33. Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then
|A| = 0.
6. If a scalar multiple of one row (or column) of A is added to
another row (or column), then the determinant does not
change.

34. Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then
|A| = 0.
6. If a scalar multiple of one row (or column) of A is added to
another row (or column), then the determinant does not
change.
7. The determinant of the product of two matrices is the product
of the determinants of those matrices:
|AB| = |A| |B|

35. Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then
|A| = 0.
6. If a scalar multiple of one row (or column) of A is added to
another row (or column), then the determinant does not
change.
7. The determinant of the product of two matrices is the product
of the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then

36. Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then
|A| = 0.
6. If a scalar multiple of one row (or column) of A is added to
another row (or column), then the determinant does not
change.
7. The determinant of the product of two matrices is the product
of the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then |αA| =

37. Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then
|A| = 0.
6. If a scalar multiple of one row (or column) of A is added to
another row (or column), then the determinant does not
change.
7. The determinant of the product of two matrices is the product
of the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then |αA| = αn |A|.