Linear Algebra: A, B, and C are pxp nonsing. matrixes. Prove

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Hi... this time i am only stuck on one problem :-/...

A, B, and C are p * p nonsingular matrices. Prove that ABC is nonsingular and that (ABC)^ (-1) = C^(-1)*B^(-1)*A^(-1).

Nonsingular means that it has an inverse. Also, because A, B, and C are square, then ABC will be square. We also know that XABC = I.

I'm thinking I should try A, A^-1, B, B^-1, etc. I'm not sure. How do I do this? Thanks
 
Do you have the theorem that states that if each of A and B is a non-singular matrix of the same order then (A*B) is non-singular and (A*B)<SUP>-1</SUP>=(B<SUP>-1</SUP>)(A<SUP>-1</SUP>).
If you do have that theorem the apply it the these two non-singular matrices (A*B) & C.
 
Can I do this?

(ABC)(C^-1 *B^-1 * C^-1) = ABC C^-1B^-1A^-1 = identity matrix. Thus, it equals C^-1B^-1A^-1?

How do i prove that its nonsingular?

And is my first part even right?

THANK YOU!
 
Because each of A and B is non-singular then (AB) is non-singular.
So (AB)C is non-singular.
Moreover, \(\displaystyle \left[ {\left( {AB} \right)C} \right]^{ - 1} = C^{ - 1} \left( {AB} \right)^{ - 1} = C^{ - 1} \left( {B^{ - 1} A^{ - 1} } \right).\)
 
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