802 CHAPTER 11 Systems of Equations and Inequalities Graphing Solution Enter matrices A and B into a graphing utility. Figure 15 shows the result using the Desmos Matrix Calculator. Multiplying Two Square Matrices If = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A B 2 1 0 4 and 3 1 1 2 find: (a) AB (b) BA EXAMPLE 10 Solution (a) = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ AB 2 1 0 4 3 1 1 2 5 4 4 8 (b) = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ BA 3 1 1 2 2 1 0 4 6 1 2 9 THEOREM Matrix multiplication is not commutative. Algebraic Solution (a) = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ AB 2 1 3 1 1 0 1 0 2 1 3 2 13 7 1 1 (b) = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ BA 1 0 2 1 3 2 2 1 3 1 1 0 2 1 3 5 1 6 8 1 9 2 by 3 3 by 2 2 by 2 3 by 2 2 by 3 3 by 3 Notice in Example 9 that AB is 2 by 2 and BA is 3 by 3. It is possible for both AB and BA to be defined and yet be unequal. In fact, even if A and B are both n by n matrices so that AB and BA are both defined and n by n, AB and BA will usually be unequal. Examples 9 and 10 prove that, unlike real number multiplication, matrix multiplication is not commutative. Now Work PROBLEMS 1 5 AND 17 Next, consider two of the properties of real numbers that are shared by matrices. Assuming that each product and each sum is defined, the following are true: Associative Property of Matrix Multiplication A BC AB C ( ) ( ) = Distributive Property A B C AB AC ( ) + = + Figure 15 Desmos Matrix Calculator
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