Practice Exam II
Math 2250-4
November 14, 2004
1. Consider the following matrix equation of

2 −1
 1
1
1 −5
the form Ax = b:
   
x1
2
6
1  x2  = 3
1
x3
3
(a) Compute the determinant of the coefficient matrix A above. What this tells you about
the matrix equation and its possible solutions. In particular, can you rule out any of the three
general possibilities: one solution, no solutions, or infinitiely many solutions, based on your
determinant computation? Explain. Answer: det(A) = 0
(b) Compute the reduced row echelon form of the 3 by 4 augmented matrix associated to
the matrix equation above, and use it to solve the system. Answer:


  

1 0 1 3
x1
3−t
0 1 0 0 , x2  =  0 
0 0 0 0
x3
t
2. Use the adjoint formula for the inverse matrix or Cramer’s rule, to solve the following system
of equations:
5x + 11y
7x + 16y
= 2
= 3
Answer: x = −1/3, y = 1/3
3. (a) Determine b such that the system has (i) infinitely many solutions (ii) no solusions:
x + 2y + z
3x + y + 2z
4x + 3y + 3z
= b
= 2b
= 1+b
Answer: (i) b = 1/2, (ii) b 6= 1/2
(b) Determine a, b such that the system has (i) infinitely many solutions (ii) no solutions:
x + 2y + z
5x + y + 2z
6x + 3y + bz
Answer: (i) b = 3, a = 1/3, (ii) b = 3, a 6= 1/3
1
=
a
= 3a
= 1+a


1
4. Let u =  −1  ,
1
are dependent.


2
v =  1 ,
0


1
w =  2  . State and apply a test that shows u, v, w
−1
Answer: c1 u + c2 v + c3 w = 0 ⇒ det([u v w]) = 0
5. Determine all values of x for which A−1 exists:


1 2
0
A =  2 0 −3  .
0 x
1
Answer: det(A) 6= 0 ⇒ x 6= 4/3
6. Solve for y in Au = b by Cramer’s

1
2
0
A= 3
2 −2
rule:

0
2 ,
1


x
u =  y ,
z


1
b =  0 .
−1
Answer: [x, y, z] = [0, 1/2, 0]
7. Consider the matrix

2 0
A= 1 1
0 1

1
1 
1
(a) Compute Det(A). What does this computation tell you about whether A is singular
or nonsingular? Answer: det(A) = 1
(b) Find the inverse matrix to A, using the row-operation algorithm. Remember to show
all work, as always. Hint: The correct inverse matrix has no fractions in it, which you could
deduce from your answer to part (a). Answer:


0
1 −1
2 −1 
A−1 =  −1
1 −2
2
(c) Recompute the inverse, using the adjoint formula. Show the cofactor matrix as well
as the adjoint matrix as intermediate steps, so that I can check your work.
(d) Use the inverse matrix from

2
 1
0
(b) or (c)

0 1
1 1 
1 1
Answer: [x1 , x2 , x3 ] = [0, 1, 1]
2
to solve the system
  
x1
1
x2  =  2 
x3
2
8. Let A be the matrix:

1
A= 0
2

−1 0
2 1 
1 1
(a) Compute the determinant of A and use it to determine whether A is singular or
nonsingular. Answer: det(A) = −1
(b) Does your answer to (a) let you deduce the reduced row echelon form of A? Explain.
Answer: rref(A) = I
(c) Find the inverse matrix to A. Answer:

−1 −1
A−1 =  −2 −1
4
3

1
1 
−2
9. Consider for symbols a, b the matrix


1 b 0
A = a 0 b  .
0 1 1
(a) Display a determinant condition for the existence of the inverse A−1 .
det(A) 6= 0 ⇒ a 6= −1, b 6= 0
Answer:
(b) Find A−1 under the conditions in (a). Answer:


b
b −b2
1
 a −1
b 
A−1 =
b(a + 1)
−a
1
ab
10. Define

2 0
A= 1 1
0 1

3
0 ,
1


1
b =  −1  ,
0
 
x1
x = x2 
x3
(a) Display the formulas for Cramer’s rule for the system Ax = b, without evaluating the
determinants
(b) Find x2 explicitly in (a). Answer: [x1 , x2 , x3 ] = [−2/5, −3/5, 3/5]
(c) Display the adjoint of the matrix A. Answer:


1
3 −3
2
3 
adj(A) =  −1
2 −3
2
3