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**Drexel28** Where the determinant came from? Behind the scenes the determinant came from noticing that if $\displaystyle T$ is an endomorphism on some $\displaystyle n$-dimensional $\displaystyle F$-space $\displaystyle V$ and $\displaystyle K:V\boxplus\cdots\boxplus V\to F$ is some alternating $\displaystyle n$-linear form then the function $\displaystyle (T\odot K)(v_1,\cdots,v_n)=K(T(v_1),\cdots,T(v_n))$ is an alternating $\displaystyle n$-linear form. But, we know that the space of alternating $\displaystyle n$-linear forms has dimension $\displaystyle \displaystyle {n\choose n}=1$. In particular $\displaystyle T\odot K=\delta_T K$ for some constant $\displaystyle \delta_T$. We then define $\displaystyle \delta_T$ to be the *determinant* of $\displaystyle T$. It's useful because it gives quantatative measure of invertibility. Namely, $\displaystyle T\in\text{GL}\left(V\right)$ if and only if $\displaystyle \delta_T\ne 0$ (or more generally if dealing with commutative rings if $\displaystyle \delta_T$ is a unit). Does that at least give you start?