798 CHAPTER 11 Systems of Equations and Inequalities A matrix whose entries are all equal to 0 is called a zero matrix . Each of the following matrices is a zero matrix. 0 0 0 0 0 0 0 0 0 0 0 0 0 [ ] ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Zero matrices have properties similar to the real number 0. If A is an m by n matrix and 0 is the m by n zero matrix, then Figure 13 Matrix operations 2 by 2 square zero matrix 2 by 3 zero matrix 1 by 3 zero matrix A A A 0 0 + = + = In other words, a zero matrix is the additive identity in matrix algebra. 2 Find Scalar Multiples of a Matrix We can also multiply a matrix by a real number. If k is a real number and A is an m by n matrix, the matrix kA is the m by n matrix formed by multiplying each entry aij in A by k. The number k is sometimes referred to as a scalar , and the matrix kA is called a scalar multiple of A. Operations Using Matrices Suppose that A B C 3 1 5 2 0 6 4 1 0 8 1 3 9 0 3 6 = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Find: (a) A4 (b) C 1 3 (c) A B 3 2 − EXAMPLE 5 Algebraic Solution Graphing Solution Enter the matrices A B , , [ ] [ ] and C[ ] into a graphing utility. Figure 13 shows the required computations using the Desmos Matrix Calculator. (a) ( ) = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⋅ ⋅ ⋅ − ⋅ ⋅ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A4 4 3 1 5 2 0 6 4 3 4 1 4 5 4 2 4 0 4 6 12 4 20 8 0 24 (b) ( ) = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⋅ ⋅ − ⋅ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ C 1 3 1 3 9 0 3 6 1 3 9 1 3 0 1 3 3 1 3 6 3 0 1 2 (c) A B 3 2 3 3 1 5 2 0 6 2 4 1 0 8 1 3 3 3 3 1 3 5 3 2 3 0 3 6 2 4 2 1 2 0 2 8 2 1 2 3 9 3 15 6 0 18 8 2 0 16 2 6 9 8 3 2 15 0 6 16 0 2 18 6 1 1 15 22 2 24 ( ) ( ) ( ) − = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⋅ ⋅ ⋅ − ⋅ ⋅ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − ⋅ ⋅ ⋅ ⋅ ⋅ − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − − − − − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Now Work PROBLEM 1 3
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