AEM_3e_Chapter_08

AEM_3e_Chapter_08 - 8 8 Matrices EXERCISES 8.1 Matrix...

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Unformatted text preview: 8 8 Matrices EXERCISES 8.1 Matrix Algebra 3. 3 × 3 6. 8 × 1 9. Not equal 12. Solving x 2 = 9, y = 4 x we obtain x = 3, y = 12 and x = − 3, t = − 12. 15. (a) A + B = µ 4 − 2 5 + 6 − 6 + 8 9 − 10 ¶ = µ 2 11 2 − 1 ¶ (b) B − A = µ − 2 − 4 6 − 5 8 + 6 − 10 − 9 ¶ = µ − 6 1 14 − 19 ¶ (c) 2 A + 3 B = µ 8 10 − 12 18 ¶ + µ − 6 1 8 24 − 30 ¶ = µ 2 2 8 12 − 12 ¶ 18. (a) AB = − 4 + 4 6 − 12 − 3 + 8 − 20 + 10 30 − 30 − 15 + 20 − 32 + 12 48 − 36 − 24 + 24 = − 6 5 − 10 0 5 − 20 12 0 (b) BA = µ − 4 + 30 − 24 − 16 + 60 − 36 1 − 15 + 16 4 − 30 + 24 ¶ = µ 2 8 2 − 2 ¶ 21. (a) A T A = (4 8 − 10) 4 8 − 10 = (180) (b) B T B = 2 4 5 (2 4 5) = 4 8 10 8 16 20 10 20 25 (c) A + B T = 4 8 − 10 + 2 4 5 = 6 12 − 5 24. (a) A T + B = µ 5 − 4 9 6 ¶ + µ − 3 11 − 7 2 ¶ = µ 2 7 2 8 ¶ (b) 2 A + B T = µ 10 18 − 8 12 ¶ + µ − 3 − 7 11 2 ¶ = µ 7 11 3 14 ¶ 113 8.1 Matrix Algebra 27. µ − 19 18 ¶ − µ 19 20 ¶ = µ − 38 − 2 ¶ 30. 3 × 2 33. ( AB ) T = µ 16 40 − 8 − 20 ¶ T = µ 16 − 8 40 − 20 ¶ ; B T A T = µ 4 2 10 5 ¶µ 2 − 3 4 2 ¶ = µ 16 − 8 40 − 20 ¶ 36. Using the property ( AB ) T = B T A T given in Problem 33 we have ( AA T ) T = ( A T ) T A T = AA T , so that AA T is symmetric. 39. Since ( A + B ) 2 = ( A + B )( A + B ) 6 = A 2 + AB + BA + B 2 , and AB 6 = BA in general, ( A + B ) 2 6 = A 2 +2 AB + B 2 . 42. 2 6 1 1 2 − 1 5 7 − 4 x 1 x 2 x 3 = 7 − 1 9 45. (a) M Y x y z = cos γ sin γ − sin γ cos γ 1 x y z = x cos γ + y sin γ − x sin γ + y cos γ z = x Y y Y z Y (b) M R = cos β − sin β 1 sin β cos β ; M P 1 cos α sin α − sin α cos α (c) M P 1 1 1 = 1 cos30 ◦ sin30 ◦ − sin30 ◦ cos30 ◦ 1 1 1 = 1 √ 3 2 1 2 − 1 2 √ 3 2 1 1 1 = 1 1 2 ( √ 3 + 1) 1 2 ( √ 3 − 1) M R M P 1 1 1 = cos45 ◦ − sin45 ◦ 1 sin45 ◦ cos45 ◦ 1 1 2 ( √ 3 + 1) 1 2 ( √ 3 − 1) = √ 2 2 − √ 2 2 1 √ 2 2 √ 2 2 1 1 2 ( √ 3 + 1) 1 2 ( √ 3 − 1) = 1 4 (3 √ 2 − √ 6) 1 2 ( √ 3 + 1) 1 4 ( √ 2 + √ 6) M Y M R M P 1 1 1 = cos60 ◦ sin60 ◦ − sin60 ◦ cos60 ◦ 1 1 4 (3 √ 2 − √ 6) 1 2 ( √ 3 + 1) 1 4 ( √ 2 + √ 6)...
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This note was uploaded on 01/03/2011 for the course BIS 511 at Yale.

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AEM_3e_Chapter_08 - 8 8 Matrices EXERCISES 8.1 Matrix...

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