Brandenburg Technical University (BTU) Cottbus
Chair of Mathematics for Engineering Prof. Dr. R. Reemtsen, Dr. F. Kemm
Mathematics for Engineering I
Problem Sheet No. 11, January 14/15, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm
Class Problems
1. Solve the following inhomogeneous system of linear equations:
1 2 1 2 4 1 2 3 4 2
.
2. Compute the LU factorization of A and find the solution of A~x=~b.
a) A=
2 4
−2 1
, ~b= 2
3
b) A=
−2 1 2 4 1 −2
−6 −3 4
, ~b=
5
−3 1
3. Find the determinant of the following matrices directly.
a) A=
1 2 1 1 3 0 1 0 1
, b) A=
2 3 1 4
.
4. Compute the determinant of the following matrix by Laplace expansion:
3 2 0 4 0
0 4 3 2 2
0 0 1 0 0
0 3 −3 1 0
−4 1 0 −2 0
.
5. Compute the determinant of the following matrix:
1 1 1 1 1
2 3 3 3 3
3 5 6 6 6
4 7 9 10 10 5 9 12 14 15
.
Homework
1. Solve the following inhomogeneous system of linear equations:
1 2 1 2 1 5 1 2 3 4 5 3
.
2. Compute the LU factorization of A and find the solution of A~x=~b.
a) A=
1 2 2 2 1 1 1 1 0 3 3 0 1 6 3 2
, ~b=
0
−1 1 1
b) A=
1 1 1 3 5 6
−2 2 7
, ~b=
2 1
−1
3. Find the determinant of the following matrices directly.
a) A=
2 1 1 2 3 0 1 0 2
, b) A=
−4 3 7 −2
.
4. Compute the determinant of the following matrix by Laplace expansion:
0 2 0 4 0
1 4 3 2 1
0 5 0 0 0
2 3 −3 3 0
−4 1 5 2 0
.
5. Compute the determinant of the following matrix:
1 2 3 4 5 2 3 4 5 1 3 4 5 1 2 4 5 1 2 3 5 1 2 3 4
.
6. For the ambitious:
In the four-dimensional space compute the parameter form of the plane that is perpendicular to the vectors (1,2,1,2)T and (1,2,3,4)T and runs trough the point (5,0,−1,0).
7. Solve the following equations and inequalities:
√1−x+√
x−2 =√
x−1, 2x2−1
x+ 2 <2x−3.
