Obligatory "ignore this space" : https://sacoronavirus.co.za

We place variables on a grid: we have an n×m matrix.

We apply a linear transformation to the identity matrix and get the matrix representation of that transformation.

We must therefore look at how a transformation is applied to a matrix, firstly; and are now only interested in square matrices.

The transformation is applied to each column, which are considered as vectors; but the results are placed as row vectors.

This transposition must be taken on faith; or we must give up on matrices altogether.

Once we have our matrix representation, we come to the discovery of the matrix transformation.

Applying this to the identity will demonstrate the necessity of transposition.

We may now consider a scalar valued function defined as the sum across a row, or down a column, of a function on every item and the matrix obtained by removing the item's row and column.

If the function is recursively defined,

*f*(c,A) = Σ[j=1][n]{ *f*(a[i][j], A[i][j]) },

we come to the class of function which includes the determinant.

Matrices don't really exist, though, do they?