Let the following be n*n square matrices: A, B, C, D(adsbygoogle = window.adsbygoogle || []).push({});

| A B |

| C D | = | A*D - C*B |

If A*C = C*A (**).

The proof:

1) A is invertible.

A' = Inverse(A).

( I 0 ) ( A B ) ( A B )

( -C*A' I ) * ( C D ) = ( 0 -C*A'*B+D )

| A B | | A B |

| C D | = | 0 -C*A'*B+D | = |A| * | -C*A'*B+D |

= | A*D - A*C*A'B | = | A*D - C*B |

2) Otherwise

Replace A by A + epsilon*I.

Now (**) is valid for the perturbated A too.

We are in case (1) again. Take the limit epsilon -> 0.

Problem:

Can you avoid this analytic method?

Do you think is there a pure algebraic derivation?

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# Let the following be n*n square matrices

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