The document provides information about matrices and determinants. It defines key terms like square matrix, determinant, and Cramer's rule. It gives examples of how to find the determinant of a 2x2 matrix and how to use Cramer's rule to solve systems of equations. The examples show setting up the matrices, calculating the determinants, and solving for the variables.

1. SECTION 8-5
Matrices and Determinants
Tue, Apr 12

2. ESSENTIAL QUESTIONS
How do you find the determinant of a 2X2 matrix?
How do you solve systems of equations using
determinants?
Where you’ll see this:
Sports, construction, fitness
Tue, Apr 12

3. VOCABULARY
1. Square Matrix:
2. Determinant:
3. Cramer’s Rule:
Tue, Apr 12

4. VOCABULARY
1. Square Matrix: A matrix with the same number of rows
and columns
2. Determinant:
3. Cramer’s Rule:
Tue, Apr 12

5. VOCABULARY
1. Square Matrix: A matrix with the same number of rows
and columns
⎡a b ⎤
2. Determinant: In a 2X2 matrix ⎢ ⎥ , det A = ad - bc.
⎣c d ⎦
3. Cramer’s Rule:
Tue, Apr 12

6. VOCABULARY
1. Square Matrix: A matrix with the same number of rows
and columns
⎡a b ⎤
2. Determinant: In a 2X2 matrix ⎢ ⎥ , det A = ad - bc.
⎣c d ⎦
3. Cramer’s Rule: A method of using determinants of
matrices to solve systems of equations
Tue, Apr 12

7. MATRIX
m x n: m rows and n columns
Tue, Apr 12

8. MATRIX
m x n: m rows and n columns
⎡ −2 2 3 ⎤ ⎡3 −1⎤
⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤
⎣ ⎦
⎣ 9 6 −3 ⎦ ⎣5 8 ⎦
Tue, Apr 12

9. MATRIX
m x n: m rows and n columns
⎡ −2 2 3 ⎤ ⎡3 −1⎤
⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤
⎣ ⎦
⎣ 9 6 −3 ⎦ ⎣5 8 ⎦
2X3
Tue, Apr 12

10. MATRIX
m x n: m rows and n columns
⎡ −2 2 3 ⎤ ⎡3 −1⎤
⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤
⎣ ⎦
⎣ 9 6 −3 ⎦ ⎣5 8 ⎦
2X3 2X2
Tue, Apr 12

11. MATRIX
m x n: m rows and n columns
⎡ −2 2 3 ⎤ ⎡3 −1⎤
⎢ ⎥ ⎢ ⎥ ⎡4 −3 0 ⎤
⎣ ⎦
⎣ 9 6 −3 ⎦ ⎣5 8 ⎦
2X3 2X2 1X3
Tue, Apr 12

12. DETERMINANT
a b
det A = = ad − bc
c d
Tue, Apr 12

13. EXAMPLE 1
⎡0 4 ⎤
Find the determinant of ⎢ ⎥.
⎣6 7 ⎦
Tue, Apr 12

14. EXAMPLE 1
⎡0 4 ⎤
Find the determinant of ⎢ ⎥.
⎣6 7 ⎦
ad − bc
Tue, Apr 12

15. EXAMPLE 1
⎡0 4 ⎤
Find the determinant of ⎢ ⎥.
⎣6 7 ⎦
ad − bc
= 0(7) − 4(6)
Tue, Apr 12

16. EXAMPLE 1
⎡0 4 ⎤
Find the determinant of ⎢ ⎥.
⎣6 7 ⎦
ad − bc
= 0(7) − 4(6)
= 0 − 24
Tue, Apr 12

17. EXAMPLE 1
⎡0 4 ⎤
Find the determinant of ⎢ ⎥.
⎣6 7 ⎦
ad − bc
= 0(7) − 4(6)
= 0 − 24
= −24
Tue, Apr 12

18. CRAMER’S RULE
Tue, Apr 12

19. CRAMER’S RULE
1. Make sure equations look like Ax + By = C.
Tue, Apr 12

20. CRAMER’S RULE
1. Make sure equations look like Ax + By = C.
2. Make a 2X2 determinant matrix: x in 1st column, y in
2nd, call A.
Tue, Apr 12

21. CRAMER’S RULE
1. Make sure equations look like Ax + By = C.
2. Make a 2X2 determinant matrix: x in 1st column, y in
2nd, call A.
3. Make a new 2X2 determinant matrix: Replace x
column with equation answers, call Ax.
Tue, Apr 12

22. CRAMER’S RULE
1. Make sure equations look like Ax + By = C.
2. Make a 2X2 determinant matrix: x in 1st column, y in
2nd, call A.
3. Make a new 2X2 determinant matrix: Replace x
column with equation answers, call Ax.
4. Make another 2X2 determinant matrix: Replace y
column with equation answers, call Ay.
Tue, Apr 12

23. CRAMER’S RULE
Tue, Apr 12

24. CRAMER’S RULE
5. Divide Ax by A and Ay by A.
Tue, Apr 12

25. CRAMER’S RULE
5. Divide Ax by A and Ay by A.
6. Check answer and rewrite solution.
Tue, Apr 12

26. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6
⎨
⎩ x + 2 y = 11
Tue, Apr 12

27. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7
⎨ A=
⎩ x + 2 y = 11 1 2
Tue, Apr 12

28. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7
⎨ A= Ax =
⎩ x + 2 y = 11 1 2 11 2
Tue, Apr 12

29. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Tue, Apr 12

30. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Ax
x=
A
Tue, Apr 12

31. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Ax(−6)(2) − (−7)(11)
x= =
A (3)(2) − (−7)(1)
Tue, Apr 12

32. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Ax(−6)(2) − (−7)(11) −12 + 77
x= = =
A (3)(2) − (−7)(1) 6+7
Tue, Apr 12

33. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Ax(−6)(2) − (−7)(11) −12 + 77 65
x= = = =
A (3)(2) − (−7)(1) 6+7 13
Tue, Apr 12

34. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Ax(−6)(2) − (−7)(11) −12 + 77 65
x= =
(3)(2) − (−7)(1)
=
6+7
= =5
A 13
Tue, Apr 12

35. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Ax(−6)(2) − (−7)(11) −12 + 77 65
x= =
(3)(2) − (−7)(1)
=
6+7
= =5
A 13
Ay
y=
A
Tue, Apr 12

36. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Ax(−6)(2) − (−7)(11) −12 + 77 65
x= =
(3)(2) − (−7)(1)
=
6+7
= =5
A 13
Ay
(3)(11) − (−6)(1)
y= =
A (3)(2) − (−7)(1)
Tue, Apr 12

37. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Ax(−6)(2) − (−7)(11) −12 + 77 65
x= =
(3)(2) − (−7)(1)
=
6+7
= =5
A 13
(3)(11) − (−6)(1) 33 + 6
Ay
y= = =
A (3)(2) − (−7)(1) 6+7
Tue, Apr 12

38. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Ax(−6)(2) − (−7)(11) −12 + 77 65
x= =
(3)(2) − (−7)(1)
=
6+7
= =5
A 13
(3)(11) − (−6)(1) 33 + 6 39
Ay
y= = = =
A (3)(2) − (−7)(1) 6 + 7 13
Tue, Apr 12

39. EXAMPLE 2
Solve the system of equations using Cramer’s rule (matrices).
⎧3x − 7 y = −6 3 −7 −6 −7 3 −6
⎨ A= Ax = Ay =
⎩ x + 2 y = 11 1 2 11 2 1 11
Ax(−6)(2) − (−7)(11) −12 + 77 65
x= =
(3)(2) − (−7)(1)
=
6+7
= =5
A 13
(3)(11) − (−6)(1) 33 + 6 39
Ay
y= = = = =3
A (3)(2) − (−7)(1) 6 + 7 13
Tue, Apr 12

40. EXAMPLE 2
⎧3x − 7 y = −6
⎨ x = 5, y = 3
⎩ x + 2 y = 11
Tue, Apr 12

41. EXAMPLE 2
⎧3x − 7 y = −6
⎨ x = 5, y = 3
⎩ x + 2 y = 11
Check:
Tue, Apr 12

42. EXAMPLE 2
⎧3x − 7 y = −6
⎨ x = 5, y = 3
⎩ x + 2 y = 11
Check: 3(5) − 7(3) = −6
Tue, Apr 12

43. EXAMPLE 2
⎧3x − 7 y = −6
⎨ x = 5, y = 3
⎩ x + 2 y = 11
Check: 3(5) − 7(3) = −6
15 − 21 = −6
Tue, Apr 12

44. EXAMPLE 2
⎧3x − 7 y = −6
⎨ x = 5, y = 3
⎩ x + 2 y = 11
Check: 3(5) − 7(3) = −6 5 + 2(3) = 11
15 − 21 = −6
Tue, Apr 12

45. EXAMPLE 2
⎧3x − 7 y = −6
⎨ x = 5, y = 3
⎩ x + 2 y = 11
Check: 3(5) − 7(3) = −6 5 + 2(3) = 11
15 − 21 = −6 5 + 6 = 11
Tue, Apr 12

46. EXAMPLE 2
⎧3x − 7 y = −6
⎨ x = 5, y = 3
⎩ x + 2 y = 11
Check: 3(5) − 7(3) = −6 5 + 2(3) = 11
15 − 21 = −6 5 + 6 = 11
(5,3)
Tue, Apr 12

47. PROBLEM SET
Tue, Apr 12

48. PROBLEM SET
p. 356 #1-31 odd
Solve all using matrices by hand
“I’m a great believer in luck, and I find the harder I work
the more I have of it.” - Thomas Jefferson
Tue, Apr 12