Math 1080
Homework #1
Problem 1:
Let ๐ต be a 4x4 matrix to which we apply the following operations:
1. Double column 3
2. Add row 2 to row 1
3. Interchange columns 2 and 3
4. Halve row 4
5. Replace column 4 by sum of columns 1 and 3
Each of these operations can be performed by multiplying ๐ต on the left or on the right by
a specific matrix ๐ธ๐ (where ๐ stands for the operation above) Find the matrices ๐ธ๐. Then
find matrices ๐ด and ๐ถ such that the result is obtained as a product ๐ด๐ต๐ถ
Problem 2:
Consider the matrix
๐ =
1
3
[
2 −1 2
2 2 −1
−1 2 2
]
Show that Q is an orthogonal matrix. What transformation of
3
IR
does it correspond to?
(Hint: Find the vector a that is invariant under Q. Pick a vector b orthogonal to a. Find
the angle α between b and Qb. If this angle is independent of the choice of b, then Q
corresponds to a rotation about a by the angle α. Think about other possibilities.)
Problem 3:
Find the 2x2 orthogonal matrix Q that corresponds to the reflection over the line
2๐ฅ − 3๐ฆ = 0.
Problem 4:
Let ๐ข, ๐ฃ be two vectors and ๐ด = ๐ผ + ๐ข๐ฃ
๐
a matrix. Show that if ๐ด is invertible, its inverse
is the matrix ๐ด
−1 = ๐ผ + ๐ผ๐ข๐ฃ
๐
and find the scalar ๐ผ.When is ๐ด singular?
Problem 5:
(a) Compute the norms โ๐คโ1, โ๐คโ2, โ๐คโ∞ for the vector ๐ค = [
3
−1
5
]
(b) Compute the norms โ๐ดโ1, โ๐ดโ2, โ๐ดโ∞ for the matrix ๐ด = [
2 −1 1
−1 0 2
]
(c) Verify the inequalities โ๐ด๐คโ๐ ≤ โ๐ดโ๐โ๐คโ๐.
