B03. 136.130: Test #3 Solutions
1. Suppose 
. Compute 
and 
.
Solution.
2. Suppose A and B
and both 4 by 4 matrices and suppose 
and 
. Find 
and 
.
Solution. 
3. (a)
Use cofactors to compute the determinant of the matrix 
. (Hint: Use a row or a column which makes the computation
easy).
Solution (a)
Use the 2nd
column: 
.
(b)
State Cramer’s rule for systems with two unknowns and two equations and
use it to solve the system
[NO marks will be given if other methods are used.]
Solution (b). Cramer’s Rule: If the unknowns
are x and y, if the coefficient matrix of the system is A, if the free coefficients are 
and 
, and if A is
invertible, then 
and 
where we get 
by putting the
free coefficients in the column number j.
Using this we have: 
and 
.