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.