# A transpose proof

## Main Question or Discussion Point

I need help on the tranpose of a multiple of a matrix.

I need to prove: transpose(AB)=transpose(B)*tranpose(A)

Any Ideas?

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HallsofIvy
Homework Helper
What is the definition of "transpose"?

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Well, from what I know, I just switch the subscripts and interchange rows and columns. So if A=a(ij) then, transpose(A)=a(ji)

I just don't know where to go from there

Did you try writing out the summation for the ij-th entry of both sides? I think it should be clear once you do that.

HallsofIvy
Homework Helper
Recall that "transpose" is not only defined for square matrices. If A is an n by m matrix and B is an m by p matrix, so that you can multiply them, then AT is an m by n matrix and BT is a p by m matrix. If n is not equal to p, you can't multiply ATBT. But you can multiply BTAT.

mathwonk
Homework Helper
if you know what it means in terms of being the induced map, namely composition, on dual spaces, the result is just the obvious fact that f*g*(h) = f*(g*h) = f*(hog) =
hogof = (gof)*(h) = (gf)*(h), so (gf)* = f*g*.

Well, I'm only taking an intro to linear algebra course. So I've never heard of an induced map.

How do you go about proving that the determinant of a nxn matrix A is equal to the determinant of the transpose of said matrix A using Laplace's expansion?

How can you use Det(AB) =Det A x det B to help with this?

d_b
what you need to do is to do a matrix multiplication for an abitrary matrix A and matrix B, I should say a transpose multiplication of matrix A and then transpose of matrix B. Then find the multiplication of matrix AB and find the transpose of that. It should be the same and that should do it.