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12.2 Multiplication of Matrices

12.2 Multiplication of Matrices. Matrix Multiplication

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12.2 Multiplication of Matrices

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  1. 12.2 Multiplication of Matrices

  2. Matrix Multiplication The product of two matrices, Am×pand Bp×n, is the matrix AB with dimensions m × n. Any element in the ith row and jth column of this product matrix is the sum of the products of the corresponding elements of the ith row of A and the jth column of B. When you multiply matrices, they need to be conformable for multiplication. This means: # of columns in 1st matrix = # of rows in 2nd matrix Ex 1) To get each element: this is the first row, first column so we take 1st row of A × 1st column of B 3 × 2 2 × 2 match dimensions of product 3 × 2 (4)(2) + (5)(5) = 8 + 25 = 33 (4)(3) + (5)(6) = 12 + 30 = 42 (7)(2) + (2)(5) = 14 + 10 = 24 write yourself how to get this element

  3. Ex 2) Find A2 (same A from Ex 1) Wait! You can’t! So… undefined 3 × 2 3 × 2 We can solve for unknown elements in a matrix equation. Ex 3) Solve for x and y. 3x – 4 = 2 3x = 6 x = 2 12 + 5y = –3 5y = –15 y = –3

  4. The Identity Matrix The identity matrix is the equivalent to the algebraic 1. Multiplying by it does not change the original. etc. Pattern: 1’s along the diagonal & 0’s everywhere else *If the product of two matrices is I, then they are inverses of each other. You can also multiply by a 0 matrix to get an O matrix.

  5. What properties are not here?? Commutative! When we “store” information in matrices, we may have to transpose them (switch rows & columns) to make them conformable for multiplication. Ex: Per 3 Per 4 Boys Girls It’s still the same!

  6. Ex 4) A fruit stand owner packages fruit in three different ways for gift packages. Economy package, E, has 6 apples, 3 oranges and 3 pears. Standard package, S, has 5 apples, 4 oranges and 4 pears. Luxury package, L, has 6 types of each fruit. The costs are $0.50 for an apple, $1.10 for an orange and $0.80 for a pear. What is the total cost of preparing each package of fruit? Number of Items apple orange pear Cost apple orange pear If we multiply in this state… the labels don’t match up cost per fruit package per fruit cost per fruit fruit per package Package E S L 1 × 3 3 × 3 1 × 3

  7. Homework #1202 Pg 608 #1, 3, 5, 8, 15, 16, 18, 19, 20, 29, 31, 34, 41, 42

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