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Unformatted text preview: AMS210.01. Quiz 2. Solutions. Andrei Antonenko 1. Which of the following statements are correct? If yes, provide the explanation. If no, provide the coun terexample. (a) Let A,B,C be square matrices of the same size. Then if AB = AC , then B = C . Solution: It is false: If A = 0, then B and C can be any matrices, but AB = AC = 0. (b) If A is an invertible matrix, then A 2 is invertible, and its inverse is equal to ( A 1 ) 2 . Solution: True: we know that ( AB ) 1 = B 1 A 1 . If B = A , then we have ( AA ) 1 = A 1 A 1 , therefore ( A 2 ) 1 = ( A 1 ) 2 . (c) Let A be a square matrix. Then if A 2 = 0 then A = 0. Solution: False: if A = ˆ 1 1 1 1 ! , then A 2 = 0, but A 6 = 0. (d) Any homogeneous system has at least one solution. Solution: True: any homogeneous system has at least one zero solution. (e) The system with 2 equations and 3 variables always has at least one solution. Solution: False: the following system doesn’t have any solutions: ( x 1 + x 2 + x 3 = 1 x 1 + x 2 + x 3 = 0 (f) If A is not invertible, then the matrix equation AX = B has no solutions. Solution: False: the equation ˆ 0 0 0 0 ! X = ˆ 0 0 0 0 ! has infinite number of solutions. (g) For any matrix A expressions AA > and A > A are always well defined. Solution: True: assume A is m × n matrix. Then A > is n × m matrix and it is possible to multiply A by A > and A > by A . (h) If AB is well defined, then B > A > is also well defined. Solution: True: if A is n × p matrix, then B should be p × m matrix (for AB to be well defined)....
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 Spring '03
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