1. Compute the determinants of the following matrices (show your work):
(a) 
4.
136.130: Test #4 Solutions20 minutes
1. Compute the determinants of the following matrices (show your work):
(a)
Solution: 
.
(b) 
Solution: 
.
(c) 
(Hint: shortcut !)
Solution: 
because the second and the fifth columns are equal (property).
2. Find all numbers 
such that the matrix 
is invertible. Justify your answer.
Solution: 
is invertible if and only if 
. We compute 
, and this is not 0 if 
or if 
. So, is invertible if and only if or .
3. Suppose 
and 
. Find 
. Justify your answer (show your work).
Solution: 
, from where we find that 
.