# Find an expression for B inverse in terms of B

• Mar 16th 2011, 04:34 PM
KelvinScc
Find an expression for B inverse in terms of B
Hi everyone,

I am stuck on this question, because I do not know any properties for square matrices when you multiply them together. I'm not sure what the question is asking, so any insight would be helpful :)

Suppose that B is an m by m matrix that satisfies the following equation:

B^5 + B^3 + B^2 = I (identity matrix)

Find an expression for B inverse in terms of B.

Is there a rule for square matrices when you multiply them by themselves?
• Mar 16th 2011, 04:55 PM
pickslides
If you keep mulitplying both sides by $\displaystyle B^{-1}$

you should be able to isolate $\displaystyle B$ in terms of $\displaystyle B^{-1}$
• Mar 16th 2011, 04:56 PM
TheEmptySet
Quote:

Originally Posted by KelvinScc
Hi everyone,

I am stuck on this question, because I do not know any properties for square matrices when you multiply them together. I'm not sure what the question is asking, so any insight would be helpful :)

Suppose that B is an m by m matrix that satisfies the following equation:

B^5 + B^3 + B^2 = I (identity matrix)

Find an expression for B inverse in terms of B.

Is there a rule for square matrices when you multiply them by themselves?

Since B is invertable just multiply the equation by its inverse

$B^{-1}(B^5 + B^3 + B^2) = B^{-1}I$

Since matrix multiplication distributes and we know that

$B^{-1}B=I$ we are done!
• Mar 17th 2011, 05:28 AM
HallsofIvy
Or, same idea "in reverse", factor out a B:
$B^5 + B^3 + B^2= B(B^4+ B^2+ B)= (B^4+ B^2+ B)B= I$ and it should be obvious what the inverse of B is, as long as you just know the definition of "inverse".