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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