Inverse matrices=]

May 2010
1
0
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
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Hi I have an assignment question as follows
Since this is an assignment, you are expected to do it! Here are some hints.

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.

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.

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