# Why don’t non-square matrices have determinants? The determinant is just the matrix’s scale factor (i.e. the “size” of the linear transformation), and I don’t see why a rectangular matrix wouldn’t have one.

What follows is an answer that I would give to my students, if they asked it in the lecture. When you wish to generalise determinants to non-square matrices, but preserve their interpretation as “scale factors”, you have to preserve the multiplicativity of determinants: scale factors of consecutively executed transformations should multiply — otherwise why call them scale factors? Hence you perhaps wish to have, for this “extended” determinant, the property

$$\det AB = \det A \cdot \det B$$ whenever the product $$AB$$ exists.

Perhaps you would also wish this “new extended” determinant to coincide with the traditional determinant when applied to square matrices.

Alas, this is impossible: take

$$A = \begin{pmatrix} 1 & 1 \end{pmatrix}$$

and

$$B = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$

then

$$AB = \begin{pmatrix} 2\end{pmatrix}$$

and

$$BA = \begin{pmatrix} 1 &1 \\ 1&1 \end{pmatrix}$$

therefore

$$\det A \cdot \det B = \det AB = \det(2) = 2$$

and

$$\det B \cdot \det A = \det BA = \det\begin{pmatrix} 1 &1 \\ 1&1 \end{pmatrix} = 0,$$

hence

$$\det A \cdot \det B \ne \det B \cdot \det A$$