MTH 122
Exam 3 Sample - Solutions
Prof. Townsend
Chapters 12 and 16
Name _______________________________________________________
You may use your calculator, your class notes, your book, and your brain as sources of information.
You may not share information or materials with others. All telecommunications devices must be
turned off.
1) Show ALL work. Unless otherwise specified, just using the TI-89 to perform the calculations
counts as your extra point for checking your work. Please write down any calculator steps you
use so that I could repeat your steps on my own calculator. Give me something to grade.
2) Provide three (3) significant digits for all non-integers.
3) You will receive extra credit for checking your work properly.
4) All problems bear the same weight.
Chapter 12 – Complex Numbers
Some useful information:
Rectangular form is a + jb
j 2 = −1
j = −1
j3 = − j
j4 = 1
−a = j a
1) Perform the indicated operations. Write the result in rectangular form, a + bj .
Be sure to show me the problem with intermediate terms written in terms of j, not just the calculator
solution.
( −1 +
)(
)
−4 3 − −64 = ( −1 + 2 j ) ( 3 − 8 j ) = 13 + 14 j
2) Perform the indicated operations. Leave your answer in rectangular form.
−3 + 2 j − 6 j − 5
−3 + 2 j − 6 j + 5 = 2 − 4 j
(
) (
)
3) Perform the indicated operations.
2 j (6 − 5 j )(6 + 5 j )
2 j ( 36 + 25) = 2 ( 61) j = 122 j
4) Simplify the expression
⎛ 15 ⎞ ⎛ 1 ⎞
⎛ 15 ⎞ ⎛ 1 ⎞
4 − ⎜ − ⎟ ⎜ − ⎟ = 4 − j2 ⎜ ⎟ ⎜ ⎟ = 4 + 1 = 5
⎝ 3 ⎠ ⎝ 5⎠
⎝ 3 ⎠ ⎝ 5⎠
5) Simplify the complex number to its rectangular form.
4 j + 2 16 j 2 = 4 j + 2 −16 = 4 j + 2 j 16 = 12 j
6) Solve the equation for x and y.
3yj + 3 j 2 = y − xj 7 + 2x
3yj − 3 = y + xj + 2x
3y = x
−3 = y + 2x
3
9
y = − ,x = −
7
7
7) Simplify the following expression into the form a + bj
j 82 − j 9 + 2 = 1 − j
8) Perform the indicated operations. Write the result in rectangular form, a + bj .
1
1
−
(2 − j ) j
1 (2 + j ) 1 (− j ) 2 + j
2+ j
2+6j
−
=
− (− j ) =
+j=
5
5
(2 − j ) (2 + j ) j (− j ) 5
9) Perform the indicated operations. Write the result in rectangular form, a + bj . You may use only the
calculator for the calculation but if you do, there is no extra credit for checking your work. Be sure to
write down your calculator input so that if there is an error, you can earn partial credit.
( −1+ 2 j )(1− j ) = (1+ 3 j ) (11− 13 j ) = 5 + 2 j
(1+ 3 j )( 5 − 2 j ) (11+ 13 j ) (11− 13 j ) 29
10) Perform the indicated operations. You may use just your calculator or do the problem by hand.
Note, if you only state the answer and it is incorrect, you will get no credit for the problem. Hence,
write out your calculator entry to give me something to grade. Write your answer in rectangular form.
3 j − 1 3 j − 1 + 3 j3
3 j − 1 3 j − 1− 3 j
− 3 j − 1 1− 3 j
=
=
2
65
8− j j +8
1 + 64
8− j j −8j
(
)(
( )(
)
)
(
(
)(
)(
)
)
(
(
)
)
Chapter 16 - Matrices

 
11) Solve the following problem using matrices, Ax = b . Show the detailed form of A and b .
x = −y + 6
x+y=6
−5y − 2 = −4x
4x − 5y = 2
⎡ 1 1 ⎤⎡ x ⎤ ⎡ 6 ⎤ ⎡ x ⎤
1 ⎡ −5 −1 ⎤ ⎡ 6 ⎤ 1 ⎡ 32 ⎤
⎢
⎥⎢ y ⎥ = ⎢
⎥ ⎢ y ⎥=
⎢
⎥⎢
⎥= ⎢
⎥
⎥⎦ ⎣ 2 ⎦ ⎢⎣
⎥⎦ −5 − 4 ⎣ −4 1 ⎦ ⎣ 2 ⎦ 9 ⎣ 22 ⎦
⎣ 4 −5 ⎦ ⎢⎣
12) Find the inverse of the matrix A by hand showing all details. You may check your answer using the
calculator.
⎡ 1 2 ⎤
⎡ −4 −2 ⎤ 1 ⎡ −4 −2 ⎤
1
A=⎢
A −1 =
⎥
⎢
⎥= ⎢
⎥
−4 − (−6) ⎣ 3 1 ⎦ 2 ⎣ 3 1 ⎦
⎣ −3 −4 ⎦
13) Find the 21 element of the following matrix product. Show all details.
⎡ 2 1 3 −1 ⎤ ⎡ 2 −1 ⎤
⎢
⎥⎢
⎥
−2 3 −4 0 ⎥ ⎢ 3 −2 ⎥
A=⎢
a21 = (−2)(2) + 3(3) − 4(2) = −3
⎢ 0 1 −1 4 ⎥ ⎢ 2 0 ⎥
⎢ 1 2 4 −3 ⎥ ⎢ 3 1 ⎥
⎣
⎦⎣
⎦
14) Use the calculator to multiply the two matrices.
the problem by hand.
⎡ 2 1 3 −1 ⎤ ⎡ 2 −1 ⎤ ⎡ 10
⎢
⎥⎢
⎥ ⎢
−2 3 −4 0 ⎥ ⎢ 3 −2 ⎥ ⎢ −3
A=⎢
=
⎢ 0 1 −1 4 ⎥ ⎢ 2 0 ⎥ ⎢ 13
⎢ 1 2 4 −3 ⎥ ⎢ 3 1 ⎥ ⎢ 7
⎣
⎦⎣
⎦ ⎣
Write the full resulting matrix. You may also do
−5
−4
2
−8
⎤
⎥
⎥
⎥
⎥
⎦
15) Solve the following system of equations using the inverse of the coefficient matrix. Be sure to
 

show me (1) the coefficient matrix A in Ax = c , (2) the inverse matrix A-1, and (3) the vector c .
–x + y –
3x – 2y +
x -
⎡ −1 1 −1 ⎤
A = ⎢ 3 −2 1 ⎥
⎢
⎥
⎢⎣ 1 −1 0 ⎥⎦
z = 1
z = 1
y = 2
⎡ 1 ⎤
 ⎢
c= 1 ⎥
⎢
⎥
⎢⎣ 2 ⎥⎦
⎡ −1 1 −1 ⎤ ⎡ x ⎤ ⎡ 1 ⎤
⎢
⎥⎢ y ⎥ = ⎢
⎥
3
−2
1
⎥ ⎢ 1 ⎥
⎢
⎥⎢
⎢⎣ 1 −1 0 ⎥⎦ ⎢⎣ z ⎥⎦ ⎢⎣ 2 ⎥⎦
⎡ 1 1 −1 ⎤
A = ⎢ 1 1 −2 ⎥
⎢
⎥
⎢⎣ −1 0 −1 ⎥⎦
−1
⎡ x ⎤ ⎡ 1 1 −1 ⎤ ⎡ 1 ⎤ ⎡ 0 ⎤
⎢
⎥ ⎢
⎥⎢
⎥ ⎢
⎥
⎢ y ⎥ = ⎢ 1 1 −2 ⎥ ⎢ 1 ⎥ = ⎢ −2 ⎥
⎢⎣ z ⎥⎦ ⎢⎣ −1 0 −1 ⎥⎦ ⎣⎢ 2 ⎥⎦ ⎢⎣ −3 ⎦⎥