9.4 Matrices 577 Next, ag gabh hbci ic , , , + = + + = + + = + and so on. Therefore, a g b h c i d j e k f l g a h b i c j d k e l f + + + + + + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = + + + + + + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ and the commutative property holds. All five properties of a commutative group are satisfied. Thus, the set of all 2 3 × matrices under the operation of matrix addition is a commutative group. 7 Now try Exercise 57 Instructor Resources for Section 9.4 in MyLab Math • Objective-Level Videos 9.4 • Animation: Matrix Addition • Animation: Matrix Multiplication • PowerPoint Lecture Slides 9.4 • MyLab Exercises and Assignments 9.4 • Chapter 9 Projects It can also be shown that certain sets of square matrices, along with the operation of matrix multiplication, form a noncommutative group. Exercises Warm Up Exercises In Exercises 1–8, fill in the blank with an appropriate word, phrase, or symbol(s). 1. A rectangular array of real numbers, symbols, or expressions is called a(n) _______. Matrix 2. A matrix that contains the same number of rows and columns is called a(n) _______ matrix. Square 3. The number of columns in a 3 2 × matrix is _______. 2 4. The number of rows in a 4 3 × matrix is _______. 4 5. To add or subtract two matrices, the matrices must have the same _______. Dimensions 6. When multiplying a matrix by a real number, the real number is called a(n) _______. Scalar 7. Multiplication of matrices is possible only when the number of _______ of the first matrix is the same as the number of _______ of the second matrix. Columns, rows 8. The set of all 2 3 × matrices under the operation of matrix addition forms a commutative _______. Group Practice the Skills In Exercises 9–12, determine A B. + 9. = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ A B 2 3 1 5 , 1 1 4 0 1 4 5 5 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 10. = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ A B 1 6 4 3 1 7 , 8 5 1 3 2 1 7 1 5 0 3 8 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 11. A B 5 2 1 4 7 0 , 3 3 4 0 1 6 = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 2 5 5 4 8 6 − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 12. A B 2 6 3 1 6 4 3 0 5 , 1 3 1 7 2 1 2 3 8 = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ * In Exercises 13–16, determine A B. − 13. = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ A B 2 8 9 1 , 1 1 3 9 1 9 6 8− ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 14. = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ A B 15 3 1 2 6 4 , 2 1 4 3 2 7 17 2 3 5 8 3 − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 15. A B 5 1 8 6 1 5 , 6 8 10 11 3 7 = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 1 9 18 17 2 2 − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 16. A B 4 7 2 5 3 1 5 1 6 , 2 4 9 0 2 4 4 3 6 = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 6 11 7 5 5 3 9 4 0 − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ *See Instructor Answer Appendix SECTION 9.4
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