All the explanations of matrix determinant leave out the motive behind the crime. This is the worst thing that can possibly be done as everything flows from that. You can get an idea where the definition of the determinant of a matrix came from by looking at the simplest case of a 2×2 system of equations:

To solve for reduce the b position coefficient to zero

Here is the seed. The solution looks like

Which you should recognize as Cramers rule lurking in the bushes.

Now let us take it a little further. Click into the links below to first see the derivation of the determinant of a 3×3 matrix. Then clink into the derivation of the determinant of a 4×4 matrix and higher with recursion.

**Derivations**

- Derivation of 3×3 determinant – This is used as a basis to develop notation that is useful in the final proof based on the 4×4 derivation
- Matrix-Determinant.lyx – original document
- Derivation of 4×4 determinant and deriving higher order determinants using recursion
- Matrix-Determinant-4×4-and-higher.lyx – original document

**Research Links**

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