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wksht08sol

# wksht08sol - MATH 152 L1 L2 Applied Linear Algebra and...

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MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Spring 2004-05 ¥ Week 08 Worksheet Solutions : Matrix Q1. (Addition and scalar multiplication) Solve 3 A - 3 1 2 2 = - 3 2 4 - 2 . Solution 3 A = - 3 2 4 - 2 + 3 1 2 2 = 0 3 6 0 , A = 1 3 0 3 6 0 = 0 1 2 0 . Q2. (Matrix multiplication) Simplify A [(3 A - 7 B ) - (7 A + 2 B )] - ( A - 2 B ) B . Solution A [(3 A - 7 B ) - (7 A + 2 B )] - ( A - 2 B ) B = A (3 A - 7 B - 7 A - 2 B ) - ( AB - 2 B 2 ) = A ( - 4 A - 9 B ) - AB + 2 B 2 = - 4 A 2 - 9 AB - AB + 2 B 2 = - 4 A 2 - 10 AB + 2 B 2 . Q3. (Matrix equation) Let A = 1 y - y 2 x and B = x - 2 x y 2 . Find x , y such that BA = O . Solution Note that BA = x + 2 xy - 4 x 2 + xy - y 4 x + y 2 . Then BA = O is equivalent to x + 2 xy = 0 , - 4 x 2 + xy = 0 , - y = 0 , 4 x + y 2 = 0 . From the third equation, we have y = 0. Then the first equation becomes x + 0 = 0, i.e., x = 0. Conversely, if x = y = 0, then all four equations are satisfied. Hence, BA = O if and only if x = y = 0. Q4. (Inverse) Verify that A = 0 0 1 0 1 2 1 2 3 and B = 1 - 2 1 - 2 1 0 1 0 0 are inverse to each other. Then compute the inverses of A 2 and A t .

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wksht08sol - MATH 152 L1 L2 Applied Linear Algebra and...

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