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|.