1. BELL-WORK Solve the system of equations using matrices:

2. HW 4.6(d) Due tomorrow: Handout (even) Calculate the inverse

3. HW 4.6(c) Solutions 5.(2,-1) 7.(0,-5) 9.(3,-5) 12.(06,-7) 14.(3,8) 16.(2,-1) 18.(6,-6) 20.(1,1) 22.(-1,-1)

4. Guiding question: How is the inverse of a 3×3 matrix calculated?

5. Matrices We can use matrices to solve a system of 3 linear equations! Therefore, we need to be able to find the inverse of a 3×3 matrix. This is a more complicated process than for a 2×2 matrix.

6. Finding the Inverse of a 3×3 Matrix Step 1: Calculating the Matrix of Minors. Step 2: Turn that into the Matrix of Cofactors. Step 3: Turn that matrix into the Adjugate. Step 4: Multiply that by 1/Determinant of the original matrix.

7. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values.

8. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for

9. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for

10. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for

11. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for Determinant =

12. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for Determinant = 2

13. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for

14. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for

15. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for Determinant =

16. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for Determinant = 2

17. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for

18. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for

19. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for Determinant =

20. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Example: Calculate the matrix of minors for Determinant = 2

21. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Continue this process…

22. The Matrix of Minors For each element of the matrix, the matrix of minors is formed by ignoring the values on the current row and column and calculating the determinant of the remaining values. Matrix of minors =

23. The Matrix of Cofactors In the Matrix of Minors, change the sign of alternate cells as follows:

24. The Matrix of Cofactors In the Matrix of Minors, change the sign of alternate cells as follows:

25. The Matrix of Cofactors Matrix of minors =

26. The Matrix of Cofactors Matrix of cofactors =

27. The Adjugate Transpose all elements of the matrix of cofactors by swapping their positions over the diagonal (the diagonal stays the same).

28. The Adjugate Transpose all elements of the matrix of cofactors by swapping their positions over the diagonal (the diagonal stays the same). Matrix of cofactors =

29. The Adjugate Transpose all elements of the matrix of cofactors by swapping their positions over the diagonal (the diagonal stays the same). Matrix of cofactors = Adjugate =

30. The Determinant of a 3×3 Matrix is given by

31. The Determinant of a 3×3 Matrix is given by

32. The Determinant of a 3×3 Matrix is given by – + 3×2

33. The Determinant of a 3×3 Matrix is given by – + 3×2 – 0×2

34. The Determinant of a 3×3 Matrix is given by – + 3×2 – 0×2 + 2×2

35. The Determinant of a 3×3 Matrix is given by – + 3×2 – 0×2 + 2×2 = 10

36. The Inverse is given by

37. The Inverse is given by

38. The Inverse Handout (odd)

39. Who wants to answer the Guiding question? How is the inverse of a 3×3 matrix calculated?