Matrix and Its Transpose

**Pinkk** · February 1st 2011, 08:56 PM

Using row reduction, prove that the determinant of a matrix and the determinant of the matrix's transpose are equal.

This is my attempt, but I am not sure if this is actually using row reduction, and if this is the right approach, I am not totally sure on the details:

Suppose $A$ is a square matrix and suppose $E_{1},...,E_{n}$ is a sequence of elementary matrices such that $B = E_{n}...E_{1}A$ is an upper triangle matrix (I am not sure how to prove that such a sequence must always exist). Then clearly $det(B) = det(B^{T})$ since the determinant of an upper triangle matrix and the determinant of a lower triangle matrix is the product of the entries along the main diagonal, which are the same in $B$ and $B^{T}$ .

Now this is the part where I am stuck. So I know that $det(E_{n})\cdot\cdot\cdot det(E_{1})det(A) = det(A^{T})det(E_{1}^{T})\cdot\cdot\cdot det(E_{n}^{T})$ , but I don't know how to show/prove that for any elementary matrix $E$ , $det(E) = det(E^{T})$ .

Any help would be appreciated it, thank you.

**FernandoRevilla** · February 1st 2011, 10:18 PM

Originally Posted by Pinkk

... but I don't know how to show/prove that for any elementary matrix $E$ , $det(E) = det(E^{T})$

Analyze the transpose of every elemetary matrix type. Two types have the same traspose and for $E(\lambda)$ corresponding to $R_i+\lambda R_j \rightarrow R_i$ , prove $\det E(\lambda)=\det\left( E(\lambda)^T\right)$ using the adjoint method.

Fernando Revilla

**Pinkk** · February 2nd 2011, 03:23 PM

Okay, that makes sense, but apparently my approach is wrong since I can't use the fact that the determinant of the upper triangle matrix is the same as the determinant of the lower triangle matrix (haven't proven the fact that we can use expansion of minors using any row or column). I am stuck on how to go about this. A hint given was:

Suppose we do an elementary row operation on $A$ , obtaining $EA$ . Then $(EA)^{T} = A^{T}E^{T}$ . What sort of column operation does $E^{T}$ do on $A^{T}$ ?

Thanks.

edit: Nevermind, I've figured it out. Thanks again!

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