1. Have we ever seen this phenomenon before? Let’s do some quick multiplication…

2. What do we call this? 5 __= 5 the (multiplicative) identity 1 Let’s do some quick multiplication…

3. What will we call this? Identity Matrix I Let’s do some quick multiplication…

4. Identity Matrix 2x2 Identity Matrix 3x3 Identity Matrix

5. Identity Matrix AI= A IA= A Any matrix multiplied by the identity matrix will be equal to itself.

6. Identity Matrix Or

7. Inverse Matrix A * A -1 = I Any matrix multiplied by its inverse will be the identity matrix.

8. Solving Systems using Matrices

9. Matrix Equation A linear system can be written as a matrix equation AX=B Coefficient matrix Variable matrix Answer matrix

10. Ex. 1 Write as a matrix equation.

11. Suppose ax = b How do you solve for x? We cannot divide matrices, but we can multiply by the inverse. AX = B A -1 A -1 IX = A -1 B A -1 X = A -1 B Solving Matrix Equations

12. Ex. 2 Solve using matrices. x = -7 y = -4 A B A -1 X = A -1 B AX = B (-7, -4)

13. Ex. 3 Solve using matrices x = 5/7 y = 2 (5/7, 2)

14. Ex. 4 Solve using matrices x = 2 y = -1 z = -2 (2, -1, -2)

15. Ex. 5 Solve using matrices x = 4 y = -7 z = 2 (4, -7, 2)