# Inverse Matrices

• Dec 30th 2009, 07:01 PM
Inverse Matrices
I'm working through a tutorial on matrices: Pauls Online Notes : Linear Algebra - Finding Inverse Matrices

I'm confused about the following bit:

If \$\displaystyle A\$ is invertible then there are a set of elementry matrices \$\displaystyle E_k...E_2E_1\$ such that \$\displaystyle E_k...E_2E_1A=I_n\$. If we multiply both sides of this by \$\displaystyle A^{-1}\$:

\$\displaystyle E_k...E_2E_1AA^{-1}=I_nA^{-1}\Rightarrow A^{-1}=E_k...E_2E_1I_n\$

I'm confused by that statement. I don't see how to get to isolate the inverse to obtain \$\displaystyle A^{-1}=E_k...E_2E_1I_n\$. On the left side of the arrow, I have \$\displaystyle E_k...E_2E_1AA^{-1}=E_k...E_2E_1I=I_nA^{-1}\$, but I still don't see how this implies that \$\displaystyle A^{-1}=E_k...E_2E_1I_n\$ (Headbang)
• Dec 30th 2009, 07:11 PM
tonio
Quote:

I'm working through a tutorial on matrices: Pauls Online Notes : Linear Algebra - Finding Inverse Matrices

I'm confused about the following bit:

If \$\displaystyle A\$ is invertible then there are a set of elementry matrices \$\displaystyle E_k...E_2E_1\$ such that \$\displaystyle E_k...E_2E_1A=I_n\$. If we multiply both sides of this by \$\displaystyle A^{-1}\$:

\$\displaystyle E_k...E_2E_1AA^{-1}=I_nA^{-1}\Rightarrow A^{-1}=E_k...E_2E_1I_n\$

I'm confused by that statement. I don't see how to get to isolate the inverse to obtain \$\displaystyle A^{-1}=E_k...E_2E_1I_n\$. On the left side of the arrow, I have \$\displaystyle E_k...E_2E_1AA^{-1}=E_k...E_2E_1I=I_nA^{-1}\$, but I still don't see how this implies that \$\displaystyle A^{-1}=E_k...E_2E_1I_n\$ (Headbang)

For any square matrix \$\displaystyle K\$ of order n, \$\displaystyle KI_n=I_nK=K\$

Tonio
• Dec 30th 2009, 07:41 PM