Math: Derivation of Matrix Determinant

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:

(matrix{2}{2}{a b c d})(matrix{2}{1}{X_1 X_2})=(matrix{2}{1}{k_1 k_2})

To solve for X_1 reduce the b position coefficient to zero

(matrix{2}{2}{ad-cb 0 bc bd})(matrix{2}{1}{X_1 X_2})=(matrix{2}{1}{dk_1-bk_2 bk_2})

Here is the seed.  The solution looks like 

Det(matrix{2}{2}{a b c d})*X_1=Det(matrix{2}{2}{k_1b k_2d})

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

 

Research Links

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