Are determinants only meaningful over the field of the linear transformation?

Question

Suppose I have a standard matrix $[A]^B_B$ for a linear transformation $(\mathbb{F}_2)^3 \rightarrow (\mathbb{F}_2)^3$.

Is $\det([A]_B^B) \in \mathbb{F}_2$? My intuition is yes, but I cannot find any source confirming this.

When performing arithmetic and cofactor expansion, do we take the definition of addition and multiplication as over $\mathbb{F}_2$?

Answer 1

There are a bunch of different questions here.

Are determinants only meaningful over the field of the linear transformation?

No, for instance they're meaningful over any field containing it.

Is the determinant of a matrix over a field in that field?

Yes, because it's a polynomial in the entries of the matrix.

When performing arithmetic and cofactor expansion, do we take the definition of addition and multiplication as over the field?

Yes. You seem to have some problems with the basic definitions here. Maybe review what a matrix is, etc.

Answer 2

The Leibniz formula for the determinant is given by (for an $n \times n$ matrix $A$ with entries $a_{i,j}$) $$ \det(A) = \sum_{\sigma \in S_n}\operatorname{sgn}(\sigma) \left( \prod_{i=1}^n a_{i,\sigma(i)} \right) $$ where

• $S_n$ is the symmetric group of order $n$ (the bijections $\{1,2,\cdots,n\} \to \{1,2,\cdots,n\}$)
• The sign of $\sigma \in S_n$ is denoted $\operatorname{sgn}(\sigma)$, and is $(-1)^m$, where $m$ is the minimum number of transpositions it takes to return $\sigma$ to the identity permutation

One concludes that (since $n$ and $S_n$ are finite), interpreting $1$ and $-1$ as in the field of concern, that $\det(A)$ lies in the field as well.

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One can also argue that, since $A$ has characteristic polynomial $p_A(\lambda) := \det(A - \lambda I)$, for which $\det(A) = p_A(0)$, one immediately has the desired result. If $A \in M_{n \times n}(F)$, then $p_A \in F[\lambda]$ and so $p_A(0) \in F$.

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When performing arithmetic and cofactor expansion, do we take the definition of addition and multiplication as over $\mathbb{F}_2$?

Yes, and the same is true for other fields. This is by definition. Just as polynomial addition and multiplication in $F[x]$ is defined by operations in $F$ on the coefficients, the same is true for matrices and their operations over $F$.