Brandenburg Technical University (BTU) Cottbus
Chair of Mathematics for Enineering Prof. Dr. R. Reemtsen, Dr. F. Kemm
Mathematics for Engineering I
Problem Sheet No. 9, December 17/18, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm
Class Problems
1. Solve the following system of linear equations using the Gauss algorithm:
2x2+ 5x3 = 9,
−2x1+ 3x2+ 10x3 = 10,
−3x1+ 6x2+ 20x3 = 23.
2. Using the Gauss algorithm find a solution of A ~x=~bfor
A=
2 0 1 1
−1 1 0 −1
1 1 1 0
0 2 1 −1
, ~b= (1,1,2,3)T.
3. Using the Gauss algorithm find a solution of A ~x=~bi, i= 1,2 for
A=
1 2 1
1 0 −1
2 2 0
, ~b1 = (4,0,4)T, ~b2 = (4,1,1)T.
4. Let A = A(s) and ~b = ~b(t) be the following parameter dependent matrix and vector respectively:
A(s) =
1 1 2 2 1 s 2 s 1
, and ~b(t) =
1 0 t
.
a) Find all parameters s and t such that the system A~x =~b does not have a solution.
b) Find all parameterssandtsuch that the systemA~x=~bhas a unique solution.
Evaluate this solution.
5. Find the rank of the matrixAand the general solution of the homogeneous system A ~x= 0:
a) A=
1 1 1 1
0 1 0 1
1 1 1 4
0 −1 0 1
, b) A=
1 0 0 1 1 2 1 1 2 2 1 2 3 4 2 3
.
6. Check, whether the following vectors are linearly independent or linearly depen- dent:
~a =
4 2 7
, ~b=
6 5 12
, ~c=
20 15 40
.
Homework
1. Let the following system of linear equations be given:
x1+ 2x2+ 4x3+ 7x4 = 4, 2x1−x4 = 2, 4x2+ 8x3 + 15x4 =a . a) Solve the system for a= 6.
b) Solve the system for a= 7.
2. Investigate the solvability of the following linear systemA~x=~bin dependence on the parametersα, β ∈R for
A=
1 1 1 1 0 α 1 2 0
, ~b= (1,1, β)T. In case of solvability find the solution of the system.
3. Let the following matrix be given:
A:=
2 11 12 6 4 1
1 12 9 11 3 7
−2 1 −8 7 −3 13
1 5 6 3 2 0
.
Solve the system of linear equations
Ax=~b for the right-hand sides
a) ~b= (13,−10,−40,6)T, b) ~b= (25,6,−60,16)T,
c) ~b= (−878,−271,1264,−443)T,
4. Determine the rank of the following matrices:
A=
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
∈R5×5 ,
B =
−4i −5 + 7i −9−4i −4 + 20i 16 + 12i
1 i 1 +i 3 1−2i
−1 +i 2 3i 5−i 1−5i
∈C3×5 ,
5. Find the rank of the matrixAand the general solution of the homogeneous system A ~x= 0:
a) A=
1 2 3 4 1 1 0 0 2 3 3 4 1 0 0 1
, b) A=
1 1 2 2
2 1 1 2
3 2 3 4
1 0 −1 0
.
6. On the seventh problem sheet (lines and planes) there were problems involving systems of linear equations. Recompute them by using the Gauss algorithm.
7. Check, whether the following vectors are linearly independent or linearly depen- dent:
a)
~a = 4
2
, ~b= 6
12
, ~c= 15
40
.
b)
~a=
8 4 14
, ~b=
6 5 12
, ~c=
4 3 8
.
8. a) Evaluate AB for
A=
2 −1 1
−1 1 1
, B =
1 1 1 2 3 2
.
b) Evaluate A~band~bTA for
A=
2 −1 1
−1 1 1
1 3 1
, ~b=
1 2 3
.
9. Simplify as much as you can:
7 8− 3
8 + 1 + 5 8− 9
8 ; 2z−1
z+ 2 +3z+ 4
z−3 − 5z2+ 3z+ 11 z2 −z−6 .
