Matrix, Prove of matrix theorem

[nekteo]

prove that if ABC are nonsingular matrices,
A) (AB)^{T} = B^{T}A^{T}
B) (ABC)^{-1} = C^{-1}B^{-1}A^{-1}

I attempted to solve it by creating a random matrices by my self and solved it, however, my teacher demand an answer without "creating" a new matrices by our self...

 

[cristo]

Try considering the ijth element of the matrix. That is, for the first one, consider (AB)^T_{ij} and expand.

 

[nekteo]

o! i got the 1st question... Thx!
is the 2nd ques also using the same method?

Cheers!
Kenneth

 

[HallsofIvy]

prove that if ABC are nonsingular matrices,
A) (AB)^{T} = B^{T}A^{T}
B) (ABC)^{-1} = C^{-1}B^{-1}A^{-1}

I attempted to solve it by creating a random matrices by my self and solved it, however, my teacher demand an answer without "creating" a new matrices by our self...

Do you understand what your teacher was saying? If you "create a random matrix" and do the calculations for that matrix, then you have proved the statement is true for that matrix. You are asked to prove it is true for any matrix.

 

[rock.freak667]

For the 2nd one let D=(ABC)^{-1} and then go from there
mutilyply by ABC

ABCD=(ABC)^{-1}(ABC)
ABCD=I
then proceed to multiply by A^{-1} and so forth

for the first one, I think you need to use the property that if A is a mxn matrix and B is a nxs matrix then AB is mxs matrix..then you need to say what kind of matrix would A^{T} would be.(nxm)