I've got two questions which im struggling with:

1. Let and be matrixes such that , and . Prove that

2. Let be some matrix such that for all real matrxes. Prove that for

Printable View

- March 4th 2010, 02:05 AMvuze88matrixes
I've got two questions which im struggling with:

1. Let and be matrixes such that , and . Prove that

2. Let be some matrix such that for all real matrxes. Prove that for - March 4th 2010, 03:16 AMHallsofIvy
- March 4th 2010, 04:33 AMsatx
The word is "matrices" not "matrixes" (Happy)

- March 4th 2010, 01:00 PMvuze88
for the first one, how do i prove AB is invertible? do i need to prove anythin else other than the fact that its square?

for the second one, how did you know to choose those 4 "matrices" - March 5th 2010, 03:12 AMHallsofIvy
A matrix is invertible if and only if its determinant is non-zero. If A had determinant 0, then . but which has determinant 1, not 0. Therefore det(A) is not 0. Same thing for B. Then det(AB)= det(A)det(B) is non-zero.

I chose those four matrices because they are the "standard basis" for the vector space of 2 by 2 matrices. Every such matrix can be written as a linear combination of them:

so they "represent" all 2 by 2 matrices.