1. Find the inverse of matrix
Å1 i ã
A =
1 −i
by finding the matrix B such that AB = I and then double check BA = I.
2. Let A represent the matrix corresponding to rotate a 2-dimensional vector for 45 degree ( ), i.e.,
Ç å
,
given vector
Å √0 ã,
b =
2 2
find the solution to Ax = b using the following three different approaches. You should reach to the same answers for all approaches.
(a) Expand the linear equation system into two equations with two unknowns and then solve for x using elimination, substitution (what you learned in algebra class).
(b) Find x directly by using the fact that A applied one a vector is to rotate the vector for 45 degree. Think about how to reserve the transformation.
(c) Prove A is an orthogonal matrix first and use the fact to find the inverse of A, and then find x by calculating A−1x.
