hw1_key - Stat 771, Fall 2011: Homework 2 Due Wednesday,...

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Stat 771, Fall 2011: Homework 2 Due Wednesday, February 6 1. Let A = 2 1 1 2 , B = 2 - 1 - 1 2 , and c = 1 3 . (a) Find 3 B . Answer: 3 2 - 1 - 1 2 = 6 - 3 - 3 6 . (b) Find A - B . Answer: 2 1 1 2 - 2 - 1 - 1 2 = 0 2 2 0 . (c) Find AB . Answer: 2 1 1 2 ‚• 2 - 1 - 1 2 = ( 2 1 ) ± 2 - 1 ( 2 1 ) ± - 1 2 ( 1 2 ) ± 2 - 1 ( 1 2 ) ± - 1 2 = 2 × 2 + 1 × ( - 1) 2 × ( - 1) + 1 × 2 1 × 2 + 2 × ( - 1) 1 × ( - 1) + 2 × 2 = 3 0 0 3 Note that you only need to multiply one matrix by 1 3 to get the inverse of the other. (d) Find | A | . Answer: 2 × 2 - 1 × 1 = 3. (e) Find A - 1 . Answer: From the formula a b c d - 1 = 1 ad - bc d - b - c a , we get A - 1 = 2 3 - 1 3 - 1 3 2 3 . (f) Is A full rank? Why or why not? Answer: Yes, because either (a) A - 1 exists, or (b) | A | 6 = 0. (g) Let x = x 1 x 2 . Show x 0 Ax = 2( x 2 1 + x 1 x 2 + x 2 2 ) . Answer: x 0 Ax = £ x 1 x 2 / 2 1 1 2 ‚• x 1 x 2 = £ x 1 x 2 / 2 x 1 + x 2 x 1 + 2 x 2 = 2 x 2 1 + x 1 x 2 + x 2 x 1 + 2 x 2 2 = 2( x 2 1 + x 1 x 2 + x 2 2 ) . (h) Use A - 1 to solve the system of two equations in two unknowns 2 x 1 + x 2 = 1 x 1 + 2 x 2 = 3 ² . 1
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That is, solve the system Ax = c . Answer: x 1 x 2 = 2 3 - 1 3 - 1 3 2 3 ‚• 1 3 = 2 3 - 1 - 1 3 + 2 = - 1 3 5 3 . (i) Show
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This note was uploaded on 12/14/2011 for the course STA 771 taught by Professor Hanson during the Spring '10 term at South Carolina.

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hw1_key - Stat 771, Fall 2011: Homework 2 Due Wednesday,...

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