SECTION 11.4 Matrix Algebra 797 Figure 12 Matrix addition and subtraction Now Work PROBLEM 9 Adding and Subtracting Matrices Suppose that A B 2 4 8 3 0 1 2 3 and 3 4 0 1 6 8 2 0 = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Find: (a) A B + (b) A B− EXAMPLE 3 Algebraic Solution Graphing Solution Enter the matrices into a graphing utility. Name them A[ ] and B . [ ] Figure 12 shows the results of adding and subtracting A[ ] and B[ ] using a TI-84 Plus CE. (a) A B 2 4 8 3 0 1 2 3 3 4 0 1 6 8 2 0 2 3 4 4 8 0 3 1 0 6 1 8 2 2 3 0 1 8 8 2 6 9 4 3 ( ) + = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = + − + + − + + + + + ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ (b) A B 2 4 8 3 0 1 2 3 3 4 0 1 6 8 2 0 2 3 4 4 8 0 3 1 0 6 1 8 2 2 3 0 5 0 8 4 6 7 0 3 ( ) − = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − − − − − − − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Add corresponding entries. Subtract corresponding entries. Many of the algebraic properties of sums of real numbers are also true for sums of matrices. Suppose that A, B, and C are m by n matrices. Then matrix addition is commutative . That is, Matrix addition is also associative . That is, Although we do not prove these results, the proofs, as the following example illustrates, are based on the commutative and associative properties for real numbers. Demonstrating the Commutative Property of Matrix Addition ( ) ( ) ( ) ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = + − + − + + + − + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − + + + − + − + + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 3 1 4 0 7 1 2 1 5 3 4 2 1 3 2 1 1 4 5 0 3 7 4 1 2 2 3 1 1 5 4 3 0 4 7 1 2 1 5 3 4 2 3 1 4 0 7 EXAMPLE 4 Commutative Property of Matrix Addition A B B A + = + Associative Property of Matrix Addition A B C A B C ( ) ( ) + + = + +
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