[Solved] MATH307 Individual Homework12

Instructions: Read textbook pages 64 to 66 before working on the homework problems. Show all steps to get full credits.

1. Find the inverse of matrix

1 i

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.,

,

given vector

0 ,

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.

• 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).
• 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.
• Prove A is an orthogonal matrix first and use the fact to find the inverse of A, and then find x by calculating A¹x.