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Let $M$ be a matrix such that \begin{equation} M = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \end{equation} As I understand it, \begin{equation} \det(M) = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} = ad-bc =z \end{equation} So, assuming $m_{ij} \in \Bbb{R}$ then $z \in \Bbb{R}$.

But I thought we consider $\begin{vmatrix} a & b \\ c & d \\ \end{vmatrix}$ a number not a matrix? Apostol mentions "determinant functions". So now I am really confused. Might $\det(M)$ be a "determinant function" acting on $M$?

My Question: Are determinants matrices, numbers or functions?

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$\det$ is a function from $n \times n$ matrices to numbers. The determinant of a particular $n \times n$ matrix is a number.

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The determinant of any particular matrix is a number; the process of taking a determinant is a function, namely, the function which associates to each matrix its determinant.

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