# Inverse of a matrix

1. Jun 25, 2010

Let A be an invertible matrix.
Then Ax=e1 will give us the first column of the inverse of A.
Where e1 is the first column of the identity matrix.

How can we prove this fact??

2. Jun 25, 2010

### HallsofIvy

From $Ax= e_1$ and the fact that A is invertible, we have $A^{-1}Ax= x= A^{-1}e_1$. Now, can you convince yourself that any matrix times $e_1$ is the first column of the matrix? Try multiplying a few matrices times $e_1$ and see what happens:
What is
$$\begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix}\begin{bmatrix}1 \\ 0\end{bmatrix}$$

What is
$$\begin{bmatrix}a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix}\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}$$

3. Jun 25, 2010