1. 2x2 Matrices, Determinants and Inverses
Evaluating Determinants of 2x2 Matrices
Using Inverse Matrices to Solve Equations

2. Evaluating Determinants of 2x2 Matrices
When you multiply two matrices together, in the order AB or BA, and the result is the identity matrix, then matrices A and B are inverses.
Identity matrix for multiplication

3. Evaluating Determinants of 2x2 Matrices
You only have to prove ONE of these.
To show two matrices are inverses…
AB = I OR BA = I
AA-1 = I OR A-1A = I
Inverse of A
Inverse of A

4. Evaluating Determinants of 2x2 Matrices
Example 1:
Show that B is the multiplicative inverse of A.

5. Evaluating Determinants of 2x2 Matrices
Example 1:
Show that B is the multiplicative inverse of A.

6. Evaluating Determinants of 2x2 Matrices
Example 1:
Show that B is the multiplicative inverse of A.
AB = I. Therefore, B is the inverse of A and A is the inverse of B.

7. Evaluating Determinants of 2x2 Matrices
Example 1:
Show that B is the multiplicative inverse of A.
AB = I. Therefore, B is the inverse of A and A is the inverse of B.
Check by multiplying BA…answer should be the same

8. Evaluating Determinants of 2x2 Matrices
Example 1:
Show that B is the multiplicative inverse of A.
AB = I. Therefore, B is the inverse of A and A is the inverse of B.
Check by multiplying BA…answer should be the same

9. Evaluating Determinants of 2x2 Matrices
Example 2:
Show that the matrices are multiplicative inverses.

10. Evaluating Determinants of 2x2 Matrices
Example 2:
Show that the matrices are multiplicative inverses.
BA = I. Therefore, B is the inverse of A and A is the inverse of B.

11. Evaluating Determinants of 2x2 Matrices
The determinant is used to tell us if an inverse exists.
If det ≠ 0, an inverse exists.
If det = 0, no inverse exists. A Matrix with a determinant of zero is called a SINGULAR matrix

12. Evaluating Determinants of 2x2 Matrices
To calculate a determinant…

13. Evaluating Determinants of 2x2 Matrices
To calculate a determinant…
Multiply along the diagonal

14. Evaluating Determinants of 2x2 Matrices
To calculate a determinant…
Take the product of the leading diagonal, and subtract the product of the non-leading diagonal
Equation to find the determinant

15. Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.

16. Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.

17. Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.

18. Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.
det = -23
Therefore, there is an inverse.

19. Evaluating Determinants of 2x2 Matrices
Example 2: Evaluate the determinant.

20. Evaluating Determinants of 2x2 Matrices
Example 2: Evaluate the determinant.

21. Evaluating Determinants of 2x2 Matrices
Example 2: Evaluate the determinant.
det = 0
Therefore, there is no inverse.

22. Evaluating Determinants of 2x2 Matrices
How do you know if a matrix has an inverse AND what that inverse is?
Given , the inverse of A is given by:
Equation to find an inverse matrix
This is called the adjoint matrix. It is formed by interchanging elements in the leading diagonal and negating elements in the non-leading diagonal

23. Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.

24. Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
Step 1: Find det M

25. Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
Step 1: Find det M
det M = -2, the inverse of M exists.

26. Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
Step 2: Find the adjoint matrix. i.e

27. Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
Step 2: Find the adjoint matrix. i.e
Change signs

28. Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
Step 2: Find the adjoint matrix. i.e
Change signs
Adjoint of M

29. Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
Step 2: Find the adjoint matrix. i.e
Change positions
Adjoint of M

30. Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
Step 2: Find the adjoint matrix. i.e
Change positions
Adjoint of M

31. Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
Step 3: Use the equation to find the inverse.

32. Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
Step 3: Use the equation to find the inverse.

33. Evaluating Determinants of 2x2 Matrices
Example 2:
Determine whether the matrix has an inverse. If an inverse exists, find it.

34. Evaluating Determinants of 2x2 Matrices
Example 2:
Determine whether the matrix has an inverse. If an inverse exists, find it.

35. Evaluating Determinants of 2x2 Matrices
Example 2:
Determine whether the matrix has an inverse. If an inverse exists, find it.

36. Solving a matrix equation. AX = B Use X = A B