Problem Set 2
1. Determine the inverse of each of the following 4  4 matrices, if it
exists, using the method of Gauss-Jordan reduction.
1
0

(a) 1

0
1
0
1
1
0
0
0
1
0
0
1

1
  1 0  1  1
 3  1 0  1
.
(b) 
5
0
4
3


0
3
2
3
2. Find all number r such that the following matrices are nonsingular.
2

(a) 1
1
4
r
2
2
3
1 
 2 4 2


(b) 1 r 3 .
1 1 2
3.
(a) Let
4
A  0
 2
2
3
0
2
1  .
1 
If possible, find a matrix B such that AB  2 A .
(b) Let
1
A  0
1
2
1
3
1
2 .
2
If possible, find a matrix B such that AB  A2  2 A .
(c) Let
A
1
1
 0
4
2
3
1
1
1  .
2
1 2 
If possible, find a matrix C such that AC  0 1 .
4 1 
1
(d) If
A
1
1
 0
1
3
1
1
0
2

1
1  and B  0

4
1
1
0
1
1 
 2
 1
Compute  AB 1 .
4. Prove the following properties of matrix.
(a) A n   A 1  .
1
n
(b) Let A, B, and A  B be nonsingular matrices. Prove that
A
1
 B 1

1
 A A  B  B .
1
(c) If v1 and v2 are solutions to the linear system Ax  b , then
w  a1v1  a2 v2 is also a solution to Ax  b , where a1  a2  1 .
(d) Let A  aij  be an n  n matrix. The trace of A, Tr ( A) , is defined as
the sum of all elements on the main diagonal of A,
Tr ( A)  a11  a 22    a nn .
Show that
(i) Tr (cA)  cTr ( A) , where c is a real number.
(ii) Tr ( A  B)  Tr ( A)  Tr ( B) .
(iii) Tr ( AB)  Tr ( BA) , where B  bij  is an n  n matrix.
(iv) Tr ( AT )  Tr ( A) .
(v) Tr ( AT A)  0 .
(vi) If Tr ( AT A)  0 , then A  0 .
2
5. Solve the system of equations:
x1  2 x2 
x3  x4  4
2 x1  3x2  4 x3  3x4  1
3x1  5 x2  5 x3  4 x4  3
.
 x1  x2  3x3  2 x4  5
x p  xh , where x p is a
Find the solutions and write it as
solution of the above linear system and xh is the solutions for the
associated homogeneous linear system.
6.
(a) Write the equivalent system of linear equations for the following
linear programming problem.
3 x1  2 x2 
x3  x4  6
x1  x2  x3  x4  8
2 x1  3 x2  x3  2 x4  10
x1  0, x2  0, x3  0, x4  0.
(b) A market research organization is studying a large group of coffee buyers
who buy a can of coffee (3 brands, A, B, and C) each week. It is found that
Brand A
Brand B
0.5
0.6
Switch to A
0.25
0.3
Switch to B
0.25
0.1
Switch to C
What is the distribution as the market is stable?
3
Brand C
0.4
0.3
0.3