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**Unformatted text preview: **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 ‚ = • 3 6 ‚ , A = 1 3 • 3 6 ‚ = • 1 2 ‚ . 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....

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