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 :)