B01. 136.130: Test #1 Solutions

1. Which of the following matrices are in Reduced Row-Echelon Form, which are in Row-Echelon Form, and which are neither in RREF nor in REF? (No need to justify your answers for this question.)

, , , .

Solution. A is in REF form, not in RREF form. B is in neither. C is in RREF (so, it is in REF too). D is in neither.

2. Solve the following system using Gaussian elimination (no marks if other methods are used).

Solution. The augmented matrix is . We so row reduction:

. We have our REF (just one step was needed). The associated system is . We now solve using back substitution. The second equation gives (obviously) . Substitute that in the first equation to get , from where we find . So, the solution of the system is ,.

2. Given , , compute, if possible:

Solutions: (a)

(b)

(c) does not exist (A is a 2x3 matrix and B is 2x2 matrix and the number of columns in A is not the same as the number of rows in B).

(c) We use the formula given in the class: .