I am currently reading Jost's Book "Riemannian Geometry and Geometric Analysis" (Sixth Edition).
In 3.3 on page 103, he writes:
From the rules of multilinear algebra, it follows easily that if $A$ is a $d\times d$-matrix, and if $f_1,\dots,f_p \in V$, then $$ \ast(Af_1\wedge\cdots\wedge Af_p)=(\operatorname{det}A)\ast(f_1\wedge\cdots\wedge f_p)$$
Here $V$ is a $d$-dimensional real vector space. I cannot help myself but thinking that this formula is wrong. It fails already in the case where $d=3$ and $p=2$. Am I getting something extremely wrong here or is this formula indeed incorrect?
My problem is, that I feel that this formula is wrong on the one hand but on the other hand this is already the sixth edition of his book, so I guess if it would be wrong indeed, somebody would have found this out already.
Thanks in advance!
Edit: I know that the statement is true whenever $d=p$, I am more curious about the case when they are not equal.