Download Assignment No. 3

Department of Aerospace Engineering
AE602 Mathematics for Aerospace Engineers
Assignment No. 3
3.1 From 𝐴𝐵 = 𝐶 find a formula for 𝐴−1 . Do the same from 𝑃𝐴 = 𝐿𝑈.
3.2 If the inverse of 𝐴2 is 𝐵, show that the inverse of 𝐴 is 𝐴𝐵. (Thus 𝐴 is invertible whenever 𝐴2
is invertible.)
3.3 Use the Gauss-Jordan method to invert
1
𝐴1 = [1
0
0 0
2 −1
0
0 0
1 1], 𝐴2 = [−1
2 −1], 𝐴3 = [0 1
0 1
0 −1
2
1 1
1
1]
1
3.4 Find the inverses (in any legal way) of
1
0 0 0 1
𝐴1 =
0 0 2 0
0 3 0 0

,
𝐴2 =
1
2
1
0
2

3
0
0
[4 0 0 0]
[
0
0
0
0
0
1

0
,
𝐴3 =
3
1
4
]
a
b
0
0
c
d
0
0
0
0
a
b
[0
0
c
d]
.
3.5 Show that for any square matrix 𝐵, 𝐴 = 𝐵 + 𝐵 𝑇 is always symmetric and 𝐾 = 𝐵 − 𝐵 𝑇 is
always skew-symmetric—which means that 𝐾 𝑇 = −𝐾. Find these matrices when 𝐵 =
1 3
[
], and write 𝐵 as the sum of a symmetric matrix and a skew-symmetric matrix.
1 1
3.6 Under what conditions on its entries is 𝐴 invertible, if
𝑎
𝐴 = [𝑑
𝑓
𝑏
𝑒
0
𝑐
0]
0
or
𝑎
𝐴 = [𝑐
0
𝑏
𝑑
0
0
0] ?
𝑒
3.7 Compute the symmetric 𝐿𝐷𝐿𝑇 factorization of
1
𝐴 = [3
5
3
12
18
5
18]
30
And
𝐴=[
𝑎
𝑏
𝑏
]
𝑑
3.8 Find the inverse of
1
1
1
4
𝐴= 1
3
1
[2
0
0
0
1
0
0
1
0
1
3
1
2
1
1
]
2
3.9 Which descriptions are correct? The solutions 𝑥 of
𝑥1
1 1 1 𝑥
0
𝐴𝑥 = [
] [ 2] = [ ]
1 0 2 𝑥
0
3
form a plane, line, point, subspace, nullspace of 𝐴, column space of 𝐴.
3.10 Compute an 𝐿𝑈 factorization for
1 2 0 1
𝐴 = [0 1 1 0 ] .
1 2 0 1
Determine a set of basic variables and a set of free variables, and find the general solution to
𝐴𝑥 = 0. What is the rank of 𝐴?
3.11 For the matrix
𝐴=[
0 1 4 0
0 2 8 0
],
Determine the echelon from 𝑈, the basic variables, the free variables and the general solution to
𝐴𝑥 = 0. Then apply elimination to 𝐴𝑥 = 𝑏, with components 𝑏1 and 𝑏2 on the right side; find the
conditions for 𝐴𝑥 = 𝑏 to be consistent (that is, to have a solution) and the general solution. What
is the rank of 𝐴?
3.12 Carry out the same steps, with 𝑏1 , 𝑏2, , 𝑏3 , 𝑏4 on the right side, for the transposed matrix
0 0
𝐴=
1 2
4 8
[0 0]
3.13 (a) Find all solutions to
2
1 2 3 4
x1
x2
𝑈𝑥 = [ 0 0 1 2 ]
x3
0 0 0 0
[ x4 ]
0
= [0 ]
0
(b) If the right side is changed from (0,0,0) to (𝑎, 𝑏, 0), what are the solutions?
3