1. Multiplying Matrices Dimensions Process Matrix Properties of Addition and Multiplication

2. Multiplying Matrices • MUST have the same number of columns in the 1st matrix as rows in the 2nd matrix. • Middle dimensions must be the same • If A = 2x3 matrix and B = 3x4 matrix then • AB = 2x3· 3x4 Since 3=3 we can multiply. • The dimensions of the product of the two matrices are the number of rows in the 1st matrix by the number of columns in the 2nd matrix. • Outside dimensions • AB = 2x3 · 3x4 So the product AB = 2x4

3. Determine if you can multiply the two matrices. If so, determine the dimensions of the product. 1. Matrix A = 2x4 and Matrix B = 4x3 2x4 · 4x3 Equal, therefore AB is possible. 2x4 · 4x3 AB will be a 2x3

4. Determine if you can multiply the two matrices. If so, determine the dimensions of the product. Matrix A = 1x4 and Matrix B = 1x4 1x4 · 1x4 Since 4 and 1 are not equal AB is not possible.

5. Process for Multiplying • Combination of multiplying and adding. • Multiply all first row by first column and add the products together. • Repeat with each row and column.

6. Process ag + bj +cm ah + bk + cn ai + bl + co dg + ej + fm dh + ek + fn di + el + fo

7. Example -1(4) + 5(6) -1(-3) + 5(8) 5(4) + 2(6) 5(-3) + 2(8) 0(4) + 4(6) 0(-3) + 4(8) Can we evaluate BA? No – 2x2 · 3x2

9. Using Calculator • Put matrices into calculator – Matrix Edit • Matrix • Arrow over • Edit • Retrieve the matrices and perform appropriate operations. • Matrix • # of matrix you want to use

11. Multi-Step Problems • Use order of operations • PEMDAS – Please Excuse My Dear Aunt Sally • Parenthesis • Exponents • Multiply / Divide (left to right) • Add / Subtract (left to right) • Can use the calculator.

12. ( ) B·(A + C) = + ( )

13. Properties of Matrices • A + B = B + A Commutative of + • A+(B+C) = (A+B)+C Associative of + • A(BC) = (AB)C Associative of x • A(B+C) = AB + AC Distributive (left) • (B+C)A = BA + CA Distributive (right) • c(AB) = (cA)B = A(cB) Scalar x • AB  BA Commutative of x Does Not Hold