Homework: determinants and
solutions of systems
1. For the system below, use determinants to decide what kind of solution is
has
7x – 9z +4y = 0
5y
– 8x + z = 1
2x
– 3y + 8z = 0
(do
not actually solve it!)
2. By inspection compute the determinant of each
of the following
a. 
b. 
c. 
d. 
3. suppose each matrix
in Problem 2 is the coefficient matrix of a system Ax
= 0. In each case what can you say about the possible solutions?
4. a. Argue (show steps) why the matrix
equation Ax = kx is the same as (A – kIn)x
= 0 where k is a scalar, A is nxn and x is nx1
b. what condition, in terms of determinants, would guarantee nontrivial
solutions?
5. a. Find non trivial solutions to Ax = 2x where
b. What is the determinant of (A – 2I2)
?
6. Find values for k so that Ax
= kx has non trivial solutions (use determinants!)
where
7. Find values for k so that Ax = kx has non trivial solutions (use determinants!)
where
