# Thread: Show that A is non-singular

1. ## Show that A is non-singular

Suppose that the nxn matrix A is column equivalent to the identity In and that E1, E2,....,Ek are the elementary matrices which perform the corresponding column operations (in order) to transform A to In. Show that A is non-singular and find $A^-1$ in terms of the E's.

I am confused with the difference betwen row equivalent and column equivalent, and what is E's?

2. Originally Posted by 450081592
Suppose that the nxn matrix A is column equivalent to the identity In and that E1, E2,....,Ek are the elementary matrices which perform the corresponding column operations (in order) to transform A to In. Show that A is non-singular and find $A^-1$ in terms of the E's.

I am confused with the difference betwen row equivalent and column equivalent, and what is E's?

Row equiv. = product of the matrix by elementary matrices from the left, column equiv. = product from the right. As simple as that, and thus:

We're given that $AE_1\cdot...\cdot E_k=I_n\,\Longrightarrow\,A^{-1}=E_1\cdot ...\cdot E_k$ ...

Tonio

3. Originally Posted by tonio
Row equiv. = product of the matrix by elementary matrices from the left, column equiv. = product from the right. As simple as that, and thus:

We're given that $AE_1\cdot...\cdot E_k=I_n\,\Longrightarrow\,A^{-1}=E_1\cdot ...\cdot E_k$ ...

Tonio
so is that it? The proof is already in the question?