Inverse matrices=]

• May 19th 2010, 05:50 AM
tkau8143
Inverse matrices=]
Hi I have an assignment question as follows

A, B and C are ntimes n matrices such that AB=I and CA=I
show B= C (i have done this part)
b) i. A and B are n times n matrices that commute. Show A squared and B squared commute
ii. Give a generalisation of this result (without proof)
c. A and B are n times n matrices and n is invetible. Shoe

(A+B) A^-1(A-B)=(A-B)A^-1(A+B)

d. A and B are n times n invertible matrices that commute. Show that A^-1 and B^-1 also commute

it's fairly urgent-any help would be much appreciated
thanks
• May 19th 2010, 06:38 AM
HallsofIvy
Quote:

Originally Posted by tkau8143
Hi I have an assignment question as follows

Since this is an assignment, you are expected to do it! Here are some hints.

Quote:

A, B and C are ntimes n matrices such that AB=I and CA=I
show B= C (i have done this part)
b) i. A and B are n times n matrices that commute. Show A squared and B squared commute
ii. Give a generalisation of this result (without proof)
I presume you have done this- it's almost trivial.

Quote:

c. A and B are n times n matrices and n is invetible. Shoe

(A+B) A^-1(A-B)=(A-B)A^-1(A+B)
I presume you mean "A is invertible".

Go ahead an multiply out left and right sides. You should get the same result. The only difference between this and elementary algebra is that you have to be careful not to commute A and B.

Quote:

d. A and B are n times n invertible matrices that commute. Show that A^-1 and B^-1 also commute

it's fairly urgent-any help would be much appreciated
thanks
This is also close to being trivial.

Look at $(AB)^{-1}$ and $(BA)^{-1}$. Of course since A and B commute, those must be equal.