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[8] 1. Solve the following system using Gauss-Jordan elimination (no marks if other methods are used).

The augmented matrix is . We use row reduction to find its row reduced echelon form:

The system associated to the last matrix is , and the only solution is x=-1, y=2 (or, the vector (-1,2)).

[8] 2. Let , and . Which of the following exist: ? Briefly justify your answer. Evaluate those of that exist.

A is a 3x3 matrix, B is 2x3 matrix: so AB does not exist.

is a 3x2 matrix, C is 2x2 matrix: so exists.

A is 3x3 matrix, C is 2x2 and is also 2x2 matrix. So, A+ does not exist.

= .

[8] 3. Is always equal to ? If your answer is "no", then find two 2x2 matrices and such that .

The answer is "no". Many examples. (Chose arbitrarily 2 matrices A and B and the probability that AB=BA is 0.)