If B²=B, prove det(B)=1 or B is singular

[hola]

B is a n*n matrix

1. Let B^2 =B. Prove that either det(B) =1 or B is singular.
2. If Transpose(B) = B^-1 , what is det(B)?


Sorry I am asking, but I can't figure them out! I'd really like to improve my linear algebra skills.
Thanks!

 

[dextercioby]

The first is really easy.

HINT:What is $\mbox{det} \ A\cdot B$ equal to...?(A,B matrices n*n).Then take A=B and recover the result u were supposed to prove.

For the second HINT:multiply to the left by B and use the "det" property u used for 1.

Daniel.

 

[hola]

I don't understand, more depth please?

 

[dextercioby]

Alright.Take two n*n matrices A and B.Each of them has a determinant.Question:what is the product of their determinants...?

$$\mbox{det} \ A\cdot B =...?$$

Daniel.

 

[hola]

det(a*b) = det(a)det(b)

So for #1, I'm still stuck.
What do I do?

 

[mathwonk]

the point is that every matrix satisfies a polynomial, and that polynomial tells you rthe eigenvaleus, which tell you whetehr it is singualr or not.


now a polynomial satisfied by a matrix such that A^2 = A would be?

 

[dextercioby]

det(a*b) = det(a)det(b)

So for #1, I'm still stuck.
What do I do?

Perfect,then u must agree that

$$\mbox{det} \ B^{2}=\left(\mbox{det} \ B\right)^{2}$$ (1)

And now apply "det" on the equation

$$B^{2}=B$$(2)

and use (1) to get a quadratic algebraic eq. in $\mbox{det} \ B$.


Daniel.

 

[hola]

dextercioby, I didn't quite get your each of your steps in the last post.

Could you help me in writing the full proof for 1 and 2?
Aaargh! I feel so frustrated. I should have take regular linear algebra instead of the honors one. I suck at proofs.


I'm really sorry, but I need to solve these problems, but can't get them. Thanks!