Question 1 (18 marks] Consider the following matrix a +1 1 a +1 0 -0-1-2 a +1 1 A= a +3 0 0 -a-3 2a -a-1 -2 4 a +2 -1 -2

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Question 1 18 Marks Consider The Following Matrix A 1 1 A 1 0 0 1 2 A 1 1 A A 3 0 0 A 3 2a A 1 2 4 A 2 1 2 1 (35.63 KiB) Viewed 69 times

Question 1 (18 marks] Consider the following matrix a +1 1 a +1 0 -0-1-2 a +1 1 A= a +3 0 0 -a-3 2a -a-1 -2 4 a +2 -1 -2a a for some real number a. (a) [10 marks) Find all the conditions for a such that the system Ax = b is consistent for all be R'. Show all your steps clearly, including the elementary row operations used. (b) (1) (2 marks) Find all the conditions on a such that the reduced row echelon form of A is (010 1 09 001 -3/20 000 1 0 0 0 0 0 Explain your answer. 0 (ii) (4 marks) For the values of a found in (b)(i), show that S = -2 2 is a basis for the row space of A. (ii) (2 marks] For the values of a found in (b)(i), find a basis for the nullspace of A.