If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix

Reduced row echelon form matrices problems and solutions

Problem 645

Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix.

Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.
 
LoadingAdd to solve later

Proof.

Because $A$ has rank $n$, we know that the $n \times n$ matrix $\rref(A)$ has $n$ non-zero rows.
This means that all $n$ rows must have a leading 1.

Each leading 1 must be in a distinct column, so we must have that each of the $n$ columns has a leading 1.
In a row echelon matrix, these leading 1s must be arranged to lie on the diagonal.

In a reduced row echelon matrix, each column with a leading 1 has 0s above and below that 1.

These restrictions mean that $\rref(A)$ must be the identity matrix.


LoadingAdd to solve later

More from my site

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

More in Linear Algebra
Linear algebra problems and solutions
If Two Matrices Have the Same Rank, Are They Row-Equivalent?

If $A, B$ have the same rank, can we conclude that they are row-equivalent? If so, then prove it. If...

Close