Mathematics for Engineering I

Brandenburg Technical University (BTU) Cottbus

Chair of Mathematics for Enineering Prof. Dr. R. Reemtsen, Dr. F. Kemm

Mathematics for Engineering I

Problem Sheet No. 8, December 10/11, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm

Class Problems

1. Given are the following (2×2)-matrices and a vector~b:

A=

1 2 3 4

, B =

−1 0 3 2

, C=

2 0 0 −2

, P =

0 1 1 0

, ~b= 3

1

.

a) Calculate AB and BA and compare the results.

b) Calculate AC and CA and compare with A.

c) Calculate AP and P A and compare with A.

d) Calculate A~b,~b^TA and~b^TA^T. e) Calculate A+B and A−B.

f) Calculate A².

2. Find all parameters λ∈R that turn A_λB =BA_λ into a true statement, with

A_λ =

2 −λ

−1 λ

, B =

1 1 1 2

. 3. The matrices A, B, C, D, E and F are given by

A=











 1 2 3 4 5













, B =













a b c d

1 2 3 4

1 1/2 1/3 1/4

2 2 4 4

1 1 3 3













, C =









1/a 1 0 2/b 1 0 3/c 0 1 4/d 0 1/2







 ,

D=

5 4 3 2 1 1 2 3 4 5

, E = 1 1 1

, F = 1 1/2

with fixed numbers a, b, c, d ∈R\ {0}. Check whether the following products are defined. When they are defined, compute them:

AA^TB , DA , EA , F AD , EE^T , E^TE .

Homework

1. Given are the following (2×2)-matrices and a vector~b:

A=

−1 1 2 −3

, B =

2 1

−1 3

, C=

1 0 0 2

, X =

0 1

−1 0

, ~b= 4

−2

.

a) Calculate AB and BA and compare the results.

b) Calculate AC and CA and compare with A.

c) Calculate AX and XA and compare with A.

d) Calculate X~b,~b^TX and~b^TX^T and compare with~b e) Calculate A+B and A−B.

f) Calculate A².

2. Find all parameters λ∈R that turn A_λ =A^T_λ into a true statement, with

A_λ =

2 −λ

−1 λ

. 3. The matrices A, B, C, D, E and F are given by

A=











 1 2 3 4 5













, B =













a b c d

1 2 3 4

1 1/2 1/3 1/4

2 2 4 4

1 1 3 3













, C =









1/a 1 0 2/b 1 0 3/c 0 1 4/d 0 1/2







 ,

D=

5 4 3 2 1 1 2 3 4 5

, E = 1 1 1

, F = 1 1/2

with fixed numbers a, b, c, d ∈R\ {0}. Check whether the following products are defined. When they are defined, compute them:

A^TA , AB , BA , DB , A^TBCE^T , AF , DD^T , D^TD .

4. Given are the following complex matrices:

A =



 1 +i 2−i

3



 , B =





a 1 1 i a 1 i i a



 , C = 1−i 2 +i 3 with a fixed complex numbera. Compute the following products:

AC , ACB , CA , CBA .

5. Solve the following equations (a6= 1, a >0,a fixed):

log_a(x) + log_a(x+ 5)−log_a(150) = 0 (x >0), (log₃(x))²−7 = 3 log₃(x²).