1. MATRICES Danny Nguyen Marissa Lally Clauberte Louis HOW TO'S: ADD, SUBTRACT, AND MULTIPLY MATRICES.

2. Subtracting Matrices The dimensions of a matrix refer to the number of rows and columns of a given matrix. # of rows x # of columns The subtraction of matrices is only allowed if matrices are the SAME size!! If the matrix doesn't have the same # of rows and columns you cannot subtract them.

3. FORBIDDEN!! 4 -7 -3 5 ERROR! You CANNOT subtract 1 x 2 matrix and a 2 x 1 matrix! They are NOT the same size You can't just flip the second matrix to make it the same either!

4. CORRECT! 1 2 3 4 5 6 2 -5 1 7 10 -1 WHAT?! 1-2 2-(-5) 3-1 4-7 5-10 6-(-1) -1 7 2 -3 -5 7

5. ADDING MATRICES 0 1 2 6 5 4 9 8 7 3 4 5 HOW TO SOLVE? (0 + 6) (1 + 5) (2 + 4) (9 + 3) (8 + 4) (7 + 5) 6 6 6 12 12 12 Both or more matrices must have the same dimensions to be able to add, if not, the operation cannot be done. For example, you cannot add a 2x3 matrix with a 3x2 matrix. 0 2 4 5 7 5 4 5 2 1 ERROR!

6. Multiplying matrices (property) To multiply an m×n matrix by an n×p matrix, the ns must be the same, and the result is an m×p matrix. Associative property: A (BC) = (AB) C Example: 4 x (3 x 2) = 24 or (2 x 4) x 3 = 24 Left distributive property: A (B + C) = AB+ AC Example: 2 x (3 + 4) = 2×3 + 2×4 Associative property (scalar): C(AB)=(cA)B=A(cB) Example: 3 x (2 x 5) = (3 x 2) x 5 = 2 x (3 x 5) Right distributive property: (A + B) C = AC+ BC Example: (2+3) x4 = 2x 4 +3 x 4

7. Multiplying matrices ( scalars) Explanation: If you are multiply a matrix by a scalar you have to multiply each entry in the matrix by the scalar. Example: 2×4=82×0=0 2×1=22×-9=-18 calculations: We call the number ("2" in this case) a scalar, so this is called "scalar multiplication".

8. Multiplying matrices (dot product) Explanation: When multiply a 1 × n matrix by an n × 1 matrix, you want to know the first row is a single row and the second is a single column. You want to: name the rows and then the column. Then the product of the row and column is formed. (1 × 1 matrix) you want to do 1st row by 1st column. ( 1, 2, 3) (7, 9, 11) = 1×7 + 2×9 + 3×11 = 58 you want to do 1st row by 2nd column (1, 2, 3) (8, 10, 12) = 1×8 + 2×10 + 3×12 = 64 DONE!!