# Proving matrix inverse properties

• Jul 17th 2010, 04:00 PM
SpiffyEh
Proving matrix inverse properties
Assume that A inR^(nxn) and without using the invertible matrix theorem, prove the following:

3.1.
Spanning Sets. If A is an n x n matrix and A^(-1) exists, then the columns of A span R^n.
3.2.
Pivot Structure. If A is an n x n matrix and Ax = b has a solution for each b inR^n, then A is invertible.

3.3.
Linear Independence. If the matrix A is invertible, then the columns of A^(1)are linearly independent.

I'm not sure how to do any of these. I don't know where to begin or anything. If someone could guide me through them that would be very helpful. Thank you
• Jul 17th 2010, 09:01 PM
Failure
Quote:

Originally Posted by SpiffyEh
Assume that A inR^(nxn) and without using the invertible matrix theorem, prove the following:

3.1.
Spanning Sets. If A is an n x n matrix and A^(-1) exists, then the columns of A span R^n.
3.2.
Pivot Structure. If A is an n x n matrix and Ax = b has a solution for each b inR^n, then A is invertible.

3.3.
Linear Independence. If the matrix A is invertible, then the columns of A^(1)are linearly independent.

I'm not sure how to do any of these. I don't know where to begin or anything. If someone could guide me through them that would be very helpful. Thank you

3.1: I suggest that you try to prove the converse: that if the columns of A do not span $\displaystyle \mathbb{R}^n$, then A cannot be invertible: because, in that case, there would exist a vector $\displaystyle y\in \mathbb{R}^n\backslash \mathrm{span}(A)$ for which there is no $\displaystyle x\in \mathbb{R}^n$ with $\displaystyle Ax = y$, which contradicts invertibility of A.

3.2 To say that Ax=b has a solution for each b in $\displaystyle \mathbb{R}^n$ amounts to the same thing as saying that the span of A is the whole of $\displaystyle \mathbb{R}^n$. So this follows directly from 3.1.

3.3. If the columns of $\displaystyle A^{-1}$ are not linearly independent, then the columns of $\displaystyle A^{-1}$ cannot span the whole of $\displaystyle \mathbb{R}^n$, hence by 3.1 $\displaystyle A^{-1}$ is not invertible, thus contradicting the invertibility of $\displaystyle A$ itself.