1. Inverse & Identity
MATRICES
Last Updated: October 12, 2005

2. Inverses and Identities
5x = 3

3. Now with Matrices ?
This is the
Identity Matrix
for 2 x 2 Matrices

4. Let’s look at another example

5. What do we multiply a matrix by to get the Identity?
New Question ?
What do we multiply a matrix by to get the Identity?

6. The Inverse of a 2x2 Matrix

7. The Inverse of a 2x2 Matrix

8. The Inverse of a 2x2 Matrix

9. A B
This is our Formula!

10. A B
This is our Formula!

11. A B C
This is our Formula!

12. Are the two Matrices Inverses?
The product of inverse matrices is the identity matrix.
Identity, therefore, INVERSE
Matrices

13. Are the two Matrices Inverses?
The product of inverse matrices is the identity matrix.
Not the Identity, therefore, NOT INVERSE
Matrices

14. Does the Matrix have an Inverse?
Let’s review the definition of the Inverse of a 2x2 Matrix

15. The Inverse of a 2x2 Matrix

16. Does the Matrix have an Inverse?
Find the determinant!
Therefore, NO inverse!

17. Does the Matrix have an Inverse?
Find the determinant!
Therefore, an inverse exists!

18. Does the Matrix have an Inverse?+ + 1 2 3 4 5 6 7 8 9
Find the determinant! - - -
1•5•9 + 2•6•7 + 3•4•8 - 7•5•3 - 8•6•1 - 9•4•2
Therefore, NO inverse!

19. Does the Matrix have an Inverse?+ + 1 3 2 4
Find the determinant! - - -
1•4•2 + 3•1•3 + 2•2•4 - 3•4•2 - 4•1•1 - 2•2•3 -7
Therefore, an inverse exists!

20. Solve the system using inverse matricesB
3x + 2y = 7
4x - 5y = 11
This is our Formula!

21. Solve the system using inverse matricesB
2x - 4y = 9
3x - 2y = 1
This is our Formula!