1. Determinants of 2 x 2 and 3 x 3 Matrices

2. Determinant of a 2 x 2 Matrixb c d d b a c -

3. Example 1

4. Determinant of a 2 x 2 Matrix3 5 4 6 3 6 4 5 - -2 =

5. Example 2

6. Determinant of a 2 x 2 Matrix-4 3 5 2 -4 2 5 3 -
-23 =

7. Example 3

8. Determinant of a 2 x 2 Matrix8 4 6 5 8 5 6 4 - 16 =

9. Example 4

10. Determinant of a 3 x 3 Matrix1 3 4 2 5 6 7

11. Determinant of a 3 x 3 Matrix1 3 4 2 5 6 7
Augment the determinant with the first two columns

12. Determinant of a 3 x 3 Matrix1 3 4 2 5 6 7
Augment the determinant with the first two columns

13. Determinant of a 3 x 3 Matrix1 3 4 2 5 6 7
Augment the determinant with the first two columns

14. Determinant of a 3 x 3 Matrix1 3 4 2 5 6 7
Augment the determinant with the first two columns

15. Determinant of a 3 x 3 Matrix1 3 4 2 5 6 7

16. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 - - -

17. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 - - -

18. Determinant of a 3 x 3 Matrix1 3 4 2 5 6 7
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

19. Determinant of a 3 x 3 Matrix1 3 4 2 5 6 7
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

20. Determinant of a 3 x 3 Matrix+ 1 3 4 2 5 6 7
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

21. Determinant of a 3 x 3 Matrix+ 1 3 4 2 5 6 7
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

22. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

23. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

24. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 -
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

25. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 -
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

26. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 - -
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

27. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 - -
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

28. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 - - -
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

29. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 - - -
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

30. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 - - -
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

31. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 - - -
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

32. Determinant of a 3 x 3 Matrix+ + 1 3 4 2 5 6 7 - - -
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
= 16

33. Example 5

34. Determinant of a 3 x 3 Matrix2 1 6 4 3 7 5 9 8
Augment the determinant with the first two columns

35. Determinant of a 3 x 3 Matrix2 1 6 4 3 7 5 9 8
Augment the determinant with the first two columns

36. Determinant of a 3 x 3 Matrix2 1 6 4 3 7 5 9 8
Augment the determinant with the first two columns

37. Determinant of a 3 x 3 Matrix2 1 6 4 3 7 5 9 8
Augment the determinant with the first two columns

38. Determinant of a 3 x 3 Matrix2 1 6 4 3 7 5 9 8
Augment the determinant with the first two columns

39. Determinant of a 3 x 3 Matrix2 1 6 4 3 7 5 9 8
Augment the determinant with the first two columns

40. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 - - -

41. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 - - -

42. Determinant of a 3 x 3 Matrix2 1 6 4 3 7 5 9 8
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

43. Determinant of a 3 x 3 Matrix2 1 6 4 3 7 5 9 8
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

44. Determinant of a 3 x 3 Matrix+ 2 1 6 4 3 7 5 9 8
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

45. Determinant of a 3 x 3 Matrix+ 2 1 6 4 3 7 5 9 8
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

46. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

47. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

48. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 -
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

49. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 -
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

50. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 - -
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

51. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 - -
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

52. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 - - -
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

53. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 - - -
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

54. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 - - -
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

55. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 - - -
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

56. Determinant of a 3 x 3 Matrix+ + 2 1 6 4 3 7 5 9 8 - - -
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
= 51

57. Find the area of the Triangle
(5, 9)
Use a Determinant
(9, 5) 1 2 5 9
(1, 2)

58. Determinant of a 3 x 3 Matrix+ + 1 2 5 9 - - -
(1•9•1 + 2•1•9 + 1•5•5 - 9•9•1 - 5•1•1 - 1•5•2)
( ) 22

59. Find the area of the Quadrilateral
(3, 9)
(9, 5)
(-3, 2)
Divide into two triangles
(1, -5)

60. Find the area of the Quadrilateral
(3, 9)
The top triangle
(9, 5)
(-3, 2) -3 2 1 3 9 5
(1, -5)

61. Find the area of the Quadrilateral
(3, 9)
The bottom triangle
(9, 5)
(-3, 2) 9 5 1 -5 -3 2
(1, -5)

62. Find the area of the Quadrilateral
(3, 9)
(9, 5)
(-3, 2)
Purple Area =
Green Area =
(1, -5)
Total Area =

63. The area of the triangle is 11. Find the value(s) of k.
Use a Determinant
(1, 3)
k+1 1 k 5 3
(k+1, 1)

64. The area of the triangle is 11. Find the value(s) of k.5 3

65. The area of the triangle is 11. Find the value(s) of k.5 3

66. “Shoe Lace” Method

67. Find the area of the Triangle
(5, 9) 1 2 5 9
(9, 5)
(1, 2)

68. Find the area of the Quadrilateral
(3, 9) -3 2 3 9 5 1 -5
(9, 5)
(-3, 2)
(1, -5)