Prove that pre-multiplying a matrix A m by the elementary matrix obtained with any matrix...
Prove that pre-multiplying a matrix by the elementary matrix obtained with any matrix elementary line transformation is the same as applying said elementary line transformation on the matrix
Answer & Explanation
Let be the standard basis row vectors, and observe the following facts about matrix multiplication:
1. gives the ith row of the matrix
2. For row vectors , we have
Combine these to get the desired result for any elementary row operations.