Description
Problem 1:
Let be a 4×4 matrix to which we apply the following operations:

Double column 3

Add row 2 to row 1

Interchange columns 2 and 3

Halve row 4

Replace column 4 by sum of columns 1 and 3
Each of these operations can be performed by multiplying on the left or on the right by a specific matrix (where stands for the operation above) Find the matrices . Then find matrices and such that the result is obtained as a product
Problem 2:
Consider the matrix

1
2
−1
2
=
[ 2
2
−1]
3
−1
2
2
Show that Q is an orthogonal matrix. What transformation of IR^{3} 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 2×2 orthogonal matrix Q that corresponds to the reflection over the line 2 − 3 = 0.
Problem 4:
Let , be two vectors and = + a matrix. Show that if is invertible, its inverse is the matrix ^{−1} = + and find the scalar .When is singular?
Problem 5:
3

Compute the norms ‖ ‖_{1}, ‖ ‖_{2}, ‖ ‖_{∞} for the vector = [−1] 5

(b) Compute the norms ‖ ‖ , ‖ ‖
2
, ‖ ‖
∞
for the matrix = [ ^{2}
−1
^{1}]
1
−1
0
2
(c) Verify the inequalities ‖ ‖
≤ ‖ ‖ ‖ ‖ .