Proving that determinants of matrices are linear transformations

Question

How do you prove that a determinant is a linear transformation using the properties: $T(cx)=cT(x)$ and $T(u+v)=T(u)+T(v)$?

For instance, if you had a $3\times3$ matrix: $(1,1,2)+(x,y,z)+(1,2,1)$, how would you prove that taking its determinant with respect to the vector $(x,y,z)$ is a linear transformation?

It is not. It is linear iff n=1 – Serge P. Jun 07 '18 at 16:08 — Serge P., Jun 07 '18 at 16:08

score 6 · Answer 1 · answered Jun 07 '18 at 16:07

6

You can't prove that since the determinant is not a linear transformation. For instance, if we are working with $n\times n$ matrices, then $\det(\lambda M)=\lambda^n\det(M)$. If $\det$ was linear, that exponent shouldn't be there, right?!

answered Jun 07 '18 at 16:07

José Carlos Santos

427,504

if all columns except 1 are held fixed then wouldn't the detA be a linear transformation of that one (vector) variable ? – Michael Lee Jun 07 '18 at 16:13
3

@MichaelLee Yes, but your question made no mention to columns. – José Carlos Santos Jun 07 '18 at 16:15

Proving that determinants of matrices are linear transformations

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