Brandenburg Technical University (BTU) Cottbus
Chair of Mathematics for Engineering Prof. Dr. R. Reemtsen, Dr. F. Kemm
Mathematics for Engineering I
Problem Sheet No. 12, January 21/22, 2008 www.math.tu-cottbus.de/˜kemm/lehre/erm
Class Problems
1. Compute the inverse and the determinant of the following matrices:
A=
1 3 2 4
, B =
1 0 0 0
1 1 0 0
1 −1 1 0
1 0 2 1
.
2. Let the matrix A and B be given by A=α
1 0 0 1
+β
0 1
−1 0
, B =γ
1 0 0 1
+δ
0 1
−1 0
, withα, β, γ, δ ∈R. a) Prove the following formulas:
A−1 = Re{ 1 α+iβ}
1 0 0 1
+ Im{ 1 α+iβ}
0 1
−1 0
, (1)
AB= Re{(α+iβ)(γ+iδ)}
1 0 0 1
+ Im{(α+iβ)(γ+iδ)}
0 1
−1 0
,
(2)
A+B = (α+γ)
1 0 0 1
+ (β+δ)
0 1
−1 0
. (3)
b) Find all pairs (α, β) for which A is an orthogonal Matrix.
3. For φ∈R, check if the following matrices are orthogonal:
A=
1 0 0
0 cos(φ) sin(φ) 0 −sin(φ) cos(φ)
, B =
cos(φ) 0 sin(φ)
0 1 0
−cos(φ) 0 sin(φ)
.
Check also the matrices A2 and B2.
4. To find an orthonormal basis of the three-dimensional space, perform the Gram- Schmidt orthonormalization procedure for the following vectors:
~a= (1,2,3)T , ~b= (2,3,1)T , ~c= (2,1,3)T, d~= (0,1,0)T .
Homework
1. Compute the inverse and the determinants of the following matrices:
A =
1 2 3 2 5 8 3 8 14
, B =
2 1 0 1
−4 −3 1 −2
2 −1 3 0
8 5 −1 5
.
2. Check, if the following matrices are orthogonal:
A= 1 2
√2 −12√
6 12√
√ 2
2 12√
6 −12√ 2
0 1 √
3
, B =
0 0 −1
−√
2 √
2 0
−√
2 −√
2 0
.
3. Let the vector~a=α(1,−2,3)T be given. For which values of α is
S =
1 0 0 0 1 0 0 0 1
−2~a~aT an orthogonal matrix?
4. By using the Gram-Schmidt orthonormalization procedure find an orthonormal basis of the space spanned by the vectors
~a= (1,0,1,0)T , ~b= (0,1,1,1)T , ~c= (1,−1,0,1)T . 5. Solve the following inhomogeneous system of linear equations:
−4 8 −5 −1 −2
−12 8 −9 3 −10
−16 8 −11 5 −14
−8 8 −7 1 −6
.
6. Compute the complex square roots of z1 =−5iand z2 =−i−2.
Compute also the cubic roots ofz3 = 8i and z4 =−216.
