Math 152, Spring 2011
Solutions Assignment #10
1. Consider the complex linear system:
x
1

2
ix
2
= 3
i

x
1
+ 3
x
2
=

2
All complex numbers appearing in your final answers of the following
questions must be in the form
a
+
bi
with
a, b
real numbers.
(a) Write the system in the form
A
x
=
b
where
A
is a matrix with
complex entries and
b
is a vector with complex coordinates.
(b) Compute det
A
.
(c) Compute
A

1
.
(d) Use the previous question to solve the original system.
(e) What MATLAB commands would enter the matrix
A
and compute
its inverse?
Solution:
(a)
A
=
1

2
i

1
3
and
b
=
3
i

2
.
(b) det
A
= 1
·
3

(

1)
·
(

2
i
) = 3

2
i
.
(c) Since det
A
= 3

2
i
= 0,
A
is invertible and the formula for
inverse of 2
×
2 matrices gives
A

1
=
1
3

2
i
3
2
i
1
1
=
3 + 2
i

3

2
i

2
3
2
i
1
1
=
3 + 2
i
13
3
2
i
1
1
and so
A

1
=
1
13
9 + 6
i

4 + 6
i
3 + 2
i
3 + 2
i
.
1
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(d) Since
A
x
=
b
and
A
is invertible, we have a unique solution to
the system:
x
=
A

1
b
=
1
13
9 + 6
i

4 + 6
i
3 + 2
i
3 + 2
i
3
i

2
=
1
13
27
i

18 + 8

12
i
9
i

6

6

4
i
=
1
13

10 + 15
i

12 + 5
i
,
that is
x
1
=

10
13
+
15
13
i
and
x
2
=

12
13
+
5
13
i
.
(e)
A = [1 2i; 1 3];
inv(A)
or
Aˆ(1)
2. Let
z
1
= 1

i
and
z
2
= 1 +
√
3
i
.
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 Spring '08
 Caddmen
 Real Numbers, Complex Numbers, Previous question, Complex number, Compute det A.

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