Description
Problem 1:
Let π΅ be a 4×4 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 π΅ on the left or on the right by
a specific matrix πΈπ (where π stands for the operation above) Find the matrices πΈπ. Then
find matrices π΄ and πΆ such that the result is obtained as a product π΄π΅πΆ
Problem 2:
Consider the matrix
π =
1
3
[
2 β1 2
2 2 β1
β1 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 Ξ± 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 Ξ±. Think about other possibilities.)
Problem 3:
Find the 2×2 orthogonal matrix Q that corresponds to the reflection over the line
2π₯ β 3π¦ = 0.
Problem 4:
Let π’, π£ be two vectors and π΄ = πΌ + π’π£
π
a matrix. Show that if π΄ is invertible, its inverse
is the matrix π΄
β1 = πΌ + πΌπ’π£
π
and find the scalar πΌ.When is π΄ singular?
Problem 5:
(a) Compute the norms βπ€β1, βπ€β2, βπ€ββ for the vector π€ = [
3
β1
5
]
(b) Compute the norms βπ΄β1, βπ΄β2, βπ΄ββ for the matrix π΄ = [
2 β1 1
β1 0 2
]
(c) Verify the inequalities βπ΄π€βπ β€ βπ΄βπβπ€βπ.