Is there a way to know if some non-sqaure matrix will have the identity matrix in the left side of its reduced row echelon form without doing the elementary row operations?
Reduced row echelon form of non sqare matrices
linear algebra
linear algebra
Is there a way to know if some non-sqaure matrix will have the identity matrix in the left side of its reduced row echelon form without doing the elementary row operations?
Best Answer
Note that it is only possible for the rref of a matrix $A$ to have the identity matrix on the left side (i.e. to be of the form $[I \quad M]$) if $A$ is $m \times n$ with $m \leq n$. That is, $A$ must be a "wide" matrix.
Given that $A$ is wide, there will be an identity matrix on the left side of the rref if and only if the first $m$ columns of $A$ are linearly independent. In other words, if and only if the matrix formed from the first $m$ columns of $A$ is invertible.
In some cases there might be shortcuts that one could use (for instance if $A$ is upper triangular), but in general the most efficient way to check that a matrix is invertible by hand is to use row operations.