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assign4_soln - Math 136 Assignment 4 Solutions 3 2-1 2 1...

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Math 136 Assignment 4 Solutions 1. Let A = 1 0 - 1 2 1 - 2 , B = 3 2 - 1 0 1 0 1 3 3 , C = 2 1 - 2 . Determine the following products or state that they are undefined. Solution: a) AB = 2 - 1 - 4 4 - 1 - 8 . b) BA is undefined since B has 3 columns but A has only 2 rows. c) AC = 4 9 d) B T C = 4 - 1 - 8 e) C T C = 9 . f) BA T = 4 10 0 1 - 2 - 1 2. If AB is a 2 × 4 matrix, then what size are the matrices A and B ? Solution: Since AB is 2 × 4, A must have 2 rows and B must have 4 columns and the number of columns of A must equal the number of rows of B . Hence A is 2 × n and B is n × 4. 1
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2 3. Determine which of the following sets are linearly independent. a) S = { (1 , - 2 , 1 , 1) , (2 , - 3 , 3 , 4) , (3 , - 6 , 4 , 5) } . Solution: Consider (0 , 0 , 0 , 0) = c 1 (1 , - 2 , 1 , 1) + c 2 (2 , - 3 , 3 , 4) + c 3 (3 , - 6 , 4 , 5) = ( c 1 + 2 c 2 + 3 c 3 , - 2 c 1 - 3 c 2 - 6 c 3 , c 1 + 3 c 2 + 4 c 3 , c 1 + 4 c 2 + 5 c 3 ) This is the homogeneous system of linear equations with coefficient matrix A = 1 2 3 - 2 - 3 - 6 1 3 4 1 4 5 .
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