B06. 136.130: Test #2 Solutions

1. (a) Suppose and . Find an elementary matrix E such that B=EA.

(b) Suppose and . Find an elementary matrix E such that B=EA.

Solution. (a) We can get B from A by interchanging the second and the third row. So, E is the matrix we get by doing exactly the same operation to the identity matrix , that is .

(b) We can get B from A by multiplying the second row by (-2). So, E is the matrix we get by doing exactly the same operation to the identity matrix , that is .

2. Solve the following system using the method of inverses (NO marks if other methods are used).

Solution. The coefficient matrix is . We find the inverse: . So, the inverse is . So, the solution of the system is

3. Compute the determinants of the following matrices (there are shortcuts for the last matrix):

,    ,    

Solution. , , (because of the 0-column).