Cool way to prove things in Linear Algebra (as long as you use the elementary operations on rows and columns).
If you apply the elementary operations on the identity matrix, it then becomes an "operation matrix" for left multiplication.
The operations that you probably already know:
E1. Multiply a row by a non-zero number;
E2. Exchange two rows;
E3. Replace the row L by the row L+k*M, where M is another row and k is a non-zero real.
Make three matrices ME1, ME2 and ME3, those matrices are obtained by applying the elementary operations on the identity matrix.
If you left multiply those matrices by any matrix A, you get the result of applying the elementary operations.
Example of proof:
Theorem: Given two square matrices of order n: A and (identity of order n).;
Apply the same (finite) elementary operations on A and I, until A becomes I;
Then I becomes the multiplicative inverse of I, A^(-1).
Proof:
Left multiply both sides by the elementary operation matrices as necessary:
By hypothesis, , therefore:
Therefore:
Pretty cool hm?
>>12062647brazzers
