Thread: Matrix Questions!

Ok, I've got 3 questions.

1) Given C and D are square matrices of order n, and inverse of C ( ) exists, expand and simplify as much as possible.

My answer turns out to be



Is this the most you can simplify it? And is the way I stated alright? Or should I do it like

2) Let u be any unit vector in , and let uu (where the T indicates that the matrix is transposed).

Prove that

a)
b)

I can do a) but I cannot do b). Can anyone help me?

3)
a) For B a square matrix, expand the product (I + B)(I − B + B^2)
b) Let = , and find and
c) Using the matrix B and the result from part (b), simplify the formula in (a).
d) Let = = for B in b).

Use the results in (c) and (a) to find .
Note: represents the identity matrix. Also recall that if , if and only if ,

Ok, for this question I can do all the way up to c but then I am lost how to do d). I mean I can do it by row reducing A and then finding the inverse that way. But the question says to use my results from a and c so I am thinking there must be abother way to do it. Can anyone help me out here?

My result for a) turns out to be I + B^3. Subsequently I find that B^3 is a 3x3 matrix with all zeros and thus my answer in c turns out to be just "I", the identity matrix.

Thanks in advance for all your help!

Originally Posted by Johnaloa

Ok, I've got 3 questions.

1) Given C and D are square matrices of order n, and inverse of C ( ) exists, expand and simplify as much as possible.

My answer turns out to be

Check that last term. It's .

Looks to me like that's all you can do.

-Dan[/quote]

Originally Posted by Johnaloa

Ok, I've got 3 questions.

1) Given C and D are square matrices of order n, and inverse of C ( ) exists, expand and simplify as much as possible.

My answer turns out to be



Is this the most you can simplify it? And is the way I stated alright? Or should I do it like As topsquark says, C^{2}D^{2} should be CDCD. Also, you can write C^{-2} for C^{-1})^2.

2) Let u be any unit vector in , and let uu (where the T indicates that the matrix is transposed).

Prove that

a)
b)

I can do a) but I cannot do b). Can anyone help me? See below.

3)
a) For B a square matrix, expand the product (I + B)(I − B + B^2) I + B^3, correct.
b) Let = , and find and B^3 = 0, correct.
c) Using the matrix B and the result from part (b), simplify the formula in (a). It becomes (I + B)(I − B + B^2) = I. You're still correct up to here.
d) Let = = for B in b).

Use the results in (c) and (a) to find . From c), I − B + B^2 is the inverse of I + B.
Note: represents the identity matrix. Also recall that if , if and only if ,

Ok, for this question I can do all the way up to c but then I am lost how to do d). I mean I can do it by row reducing A and then finding the inverse that way. But the question says to use my results from a and c so I am thinking there must be another way to do it. Can anyone help me out here?

My result for a) turns out to be I + B^3. Subsequently I find that B^3 is a 3x3 matrix with all zeros and thus my answer in c turns out to be just "I", the identity matrix.

For 2), let . Then (because u is a unit vector and so ). Therefore (you do the rest).