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This question confuses me for a reason. I read the questions and they sound to simple and to easy to answer. So maybe its something im reading wrong and not answering. Help would be greatly apreciated.

first off

Let O(n) = { A | A is an n x n matrix with A^t A = I } be the set of n by n orthogonal matrices. Show that

a) I "is in" O(n)

b) If A, B "is in" O(n), then AB "is in" O(n) and that

c) If A "is in" O(n), then A^-1 "is in" O(n)

now a) just seems so simple i just dont know how to answer somthing like that

and for b) i have

-- if A,B "is in" O(n)

AA^t = I

BB^t = I

if AA^t = I , and BB^t = I then,

AA^t = BB^t

-- show AB "is in" O(n)

AB(AB)^t = I

ABB^tA^t = I

since BB^t = AA^t

AA^tAA^t = I

therefore since AA^t = I then AA^tAA^t = I and

therefore AB = I

now does this last statement change the process of the question

(In other words, this problem asks you to show that using the operation of matrix multiplication, O(n) is a group.)

does this statement change the way i should aproach a)b)c)

thanks for you time

regards,

adam