2. The Matrix Reloaded 00011000101001001110101001110001100010100100111010100111 00011000101001001110101001110001100010100100111010100111 00011000101001001110101001110001100010100100111010100111 00010010101001001110101001110001001010100100111010100111 00011010101001001110101001110001101010100100111010100111 1101100011101010101010011111011000111010101010100111 1101100011101010101010011111011000111010101010100111 1101100011101010101010011111011000111010101010100111 1101101011101010101010011111011010111010101010100111 1101100011101010101010011111011000111010101010100111 111101010101010101010101010100111111101010101010101010101010100111 111101010101010101010101010100111111101010101010101010101010100111 Matrix 2: release date May 2003 MathScience Innovation Center B. Davis

3. The Matrix Reloaded B. Davis MathScience Innovation Center Let’s review Inverses and Identities If an expression is operated on by value x and the expression remains the same, then x is called a(n) _______________ for that operation.

4. The Matrix Reloaded B. Davis MathScience Innovation Center Let’s review Inverses and Identities If an expression is operated on by value x and the expression remains the same, then x is called a(n) _______________ for that operation. Identity For real numbers, the identity element for addition is___? For real numbers, the identity element for multiplication is___? 0 1

5. The Matrix Reloaded B. Davis MathScience Innovation Center Let’s review Inverses and Identities In matrix addition, the identity matrix size must be:____________________ The same as the addend size only zeros  In matrix addition, the identity matrix must be filled with:_______________ Example: [ 2 4 ] + [ 0 0 ] = [ 2 4 ]

6. The Matrix Reloaded B. Davis MathScience Innovation Center And Multiplication... In matrix multiplication, the identity matrix size must be:____________________ A square matrix. a diagonal of 1’s and all the rest zeros  In matrix multiplication, the identity matrix must be filled with:________________

7. The Matrix Reloaded B. Davis MathScience Innovation Center And now…Inverses In matrix addition, inverse matrices are 2 matrices that add up to an identity matrix of all zeros. In matrix addition, inverse matrices are composed elements that are the additive inverses of of elements in the original matrix. Example: [ 2 4 ] + [ -2 -4 ] = [ 0 0 ]

8. The Matrix Reloaded B. Davis MathScience Innovation Center And now… Multiplication In matrix multiplication, inverse matrices are 2 matrices whose product is an identity matrix of 0’s and 1’s. By far, the easiest way to create an inverse matrix is A -1 on your TI-83plus.

9. The Matrix Reloaded B. Davis MathScience Innovation Center Multiplicative Inverses Steps: 1. Enter your matrix using MATRIX EDIT. 2. On the home screen, use MATRIX NAME (select yours) then press x -1. Try this using [A] = [A] -1 = ___?

10. The Matrix Reloaded B. Davis MathScience Innovation Center Multiplicative Inverses Now... this using [A] = Try [A] [A] -1 = ___? * =

11. The Matrix Reloaded B. Davis MathScience Innovation Center Multiplicative Inverses Therefore, since their product is the identity matrix I 2x2, A and A -1 are called inverses. = =

12. The Matrix Reloaded B. Davis MathScience Innovation Center Time to learn how to do it without technology Let’s start with [A]= Write it down. Now, here is the rule: Where ad -bc is called the determinant. There are other rules for larger matrices, but 2 x 2 is all you need to know!

13. The Matrix Reloaded B. Davis MathScience Innovation Center without technology Let’s start with the same [A]. Write it down. Now, the determinant is _____? (Check on your calculator using Matrix Math det [A]. ) -16

14. The Matrix Reloaded B. Davis MathScience Innovation Center without technology Next, let’s find the matrix If [A] = then this new matrix is formed by switching the 2 and the -6 and then turning the 4 and the 1 negative.

15. The Matrix Reloaded B. Davis MathScience Innovation Center without technology Next, let’s find the matrix If [A] = then this new matrix is formed by switching the 2 and the -6 and then turning the 4 and the 1 negative.

16. The Matrix Reloaded B. Davis MathScience Innovation Center without technology Next, let’s find the matrix If [A] = then this new matrix is formed by switching the 2 and the -6 and then turning the 4 and the 1 negative.

17. The Matrix Reloaded B. Davis MathScience Innovation Center without technology We still are not finished! We still need to multiply by 1/det. Do you remember what the det was? -16 So, multiply 1/-16 by and that is it!

18. The Matrix Reloaded B. Davis MathScience Innovation Center without technology Therefore: =

19. The Matrix Reloaded B. Davis MathScience Innovation Center Your turn to try it ! Here is the rule: And here is your matrix [A]:

20. The Matrix Reloaded B. Davis MathScience Innovation Center Your turn to try it ! What is your determinant? What is your inverse matrix [A] -1 ? -2