# Matrix invertibilty and equivalent statements

• Mar 22nd 2013, 09:45 AM
semouey161
Matrix invertibilty and equivalent statements
Hi!

I never took linear algebra, and I'm studying it on my own for the sake of learning (MIT opencourseware, online books, etc).

I'm currently working on that a matrix A is invertible, and I can show that an equivalent statement to this is that A is row equivalent to the n x n Identity matrix. Now, I'd like to show that the n x n identity matrix has n pivot positions. How would I get that started? I'm not looking for a formal proof, but an informal proof with justifications (doesn't have to be rigorous, just the sketch of the idea).

Thanks!
• Mar 22nd 2013, 04:51 PM
Gusbob
Re: Matrix invertibilty and equivalent statements
The real reason is that the identity matrix is, by definition, in rref with a leading 1 on the diagonal and zeroes everywhere else. Therefore there is a pivot position in each column (and row), and there are n columns (and row). There are also some geometric interpretations, but I'm pretty sure it all boils down to this property at the end anyways.
• Mar 24th 2013, 08:29 PM
semouey161
Re: Matrix invertibilty and equivalent statements
Thanks! That clarified it.

My next one to clarify is to show the following are equivalent (keeping in mind that A is an invertible n X n matrix) :

The columns of A span Rn.
The columns of A form a basis of Rn.

I'm still foggy on the idea of something "spanning" something else, as is the case of columns of A span Rn, and I don't know what it means that the columns of A form a basis of Rn. However, I do imagine that it is also true that the columns of A forming the basis of Rn implies that the columns of A span Rn. In other words, the first statement <==> the second statement.

So how are these two statements logically equivalent to one another?
• Mar 25th 2013, 04:16 AM
Gusbob
Re: Matrix invertibilty and equivalent statements
Are you familiar with the invertible matrix theorem?
• Mar 25th 2013, 08:38 AM
semouey161
Re: Matrix invertibilty and equivalent statements
I'm familiarizing myself with it, but I noticed different texts show different paths to show the equivalence of those statements.
• Mar 26th 2013, 11:30 PM
semouey161
Re: Matrix invertibilty and equivalent statements
If I state that the columns of A are independent and they form a span of Rn therefore, they form a basis for Rn, is that enough or am I missing more of a justification?

I'm still foggy on the idea of span.
• Mar 27th 2013, 04:57 PM
Gusbob
Re: Matrix invertibilty and equivalent statements
That is enough, assuming you know those two facts are true.

The following applies to $\displaystyle n\times n$ matrices.
Columns of A span $\displaystyle \mathbb{R}^n \iff$ There are $\displaystyle n$ pivot rows in $\displaystyle A\iff$ There are $\displaystyle n$ pivot columns in $\displaystyle A \iff$ The columns of $\displaystyle A$ form a linearly independent set. Note that the middle equivalence about pivot rows and columns is not true in general for $\displaystyle m\times n$ matrices with $\displaystyle m\not= n$
Using what you have said in the preceding post, the columns of $\displaystyle A$ form a basis of $\displaystyle \mathbb{R}^n$. The reverse implication is trivial from the definition of basis.