5.5. Solving a System using a Matrix Inverse

One might expect that since the same row operations are used to solve a system and find an inverse that a matrix inverse can be used by itself to solve a system. There are two ways to look at an inverse. The first relates back to elementary algebra where inverse operations like multiplication and division are used to solve for an unknown variable. A matrix inverse can be used to solve for the unknown solution vector in the system :   The other way to look at an inverse is that it keeps a record of all the transformations that brought a matrix to RREF. So multiplying an inverse by any matrix/vector (given that the multiplication is valid) will replay all the transformations performed on the original matrix onto the new matrix/vector. This may seem pointless since we could just row reduce to get the solution rather than row reducing and then using , but consider the situation where you may have many vectors. This situation often arises in real-life applications of linear algebra and the latter approach is more computationally efficient.