Proving Invertibility of a Matrix: Ax=e1

[Ali Asadullah]

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??

 

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

 

[Ali Asadullah]

OMG that was too simple thank u HallsofIvy :)