Math 121, Some Practice Questions for Test 4
1. The augmented matrix
1 2 0 −3
0 1 −3 0
0 0 k−4
represents a system of equations in the vari-
ables x,y,z.
(a) For what values of kis there no solution?
(b) For what values of kis there exactly one solution?
(c) Find the solution to this system of equations if k=−2.
2. Let A=2−1 4
−3 1 0 ,B=1 3
2−1, and C=3−1 1
−2 2 0 . Find the following,
if possible, if an operation is not possible, state why it is not possible.
(a) BC (b) C B (c) 2A−2C
3. Consider the following system of equations:
x+y+ 2z= 1
2x+ 3y+ 3z=−2
3x+ 3y+ 7z= 3
.
Solve this system using the fact that the inverse of
1 1 2
2 3 3
3 3 7
is
12 −1−3
−5 1 1
−3 0 1
.
4. Consider the matrices A=
12 −1−3
−5 1 1
−3 0 1
and B=
2−0−3 2
−1 3 3 0
−2 4 −1 2
(a) Let C=AB. Determine the dimensions of C, and find c23 (you don’t need to compute the
entire product).
(b) Does D=BA exist? If so, determine the dimensions of Dand find d31 .
5. (a) Let Abe a 2 by 3 matrix, how many rows and how many columns must Bhave so that
C=AB is a 2 by 8 matrix?
(b) If Ais an mby nmatrix and Bis an nby pmatrix, what are the dimensions of the product
AB?
(c) If Ais an mby nmatrix and Bis a rby smatrix. Under what condition(s) will AB exist?
Under what condition(s) will BA exist?
