Where Do Determinants Come From?

If you do Gauss-Jordan elimination to find the inverse of \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right) you will get

\left( \begin{array}{ccc} \frac{-f h+e i}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{c h-b i}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{-c e+b f}{-c e g+b f g+c d h-a f h-b d i+a e i} \\ \frac{f g-d i}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{-c g+a i}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{c d-a f}{-c e g+b f g+c d h-a f h-b d i+a e i} \\ \frac{-e g+d h}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{b g-a h}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{-b d+a e}{-c e g+b f g+c d h-a f h-b d i+a e i} \end{array} \right)

Determinant of \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right) is -c e g+b f g+c d h-a f h-b d i+a e i. This also happens to be the denominator of every term in inverse above. So now you know where determinant comes from. It is the denominator that results when you Gaus Jordan elimination on any square matrix.

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