Can a Non-Square Matrix Have an Identity Matrix as its RREF?

Defining the Identity Matrix

To answer this question, we must look at the mathematical definition of the Identity matrix. An Identity matrix, denoted as I_n, is defined as a square matrix (having an equal number of rows and columns, n × n) with ones on the main diagonal and zeros everywhere else.

Because a non-square matrix (such as a 3x4 or 5x2 matrix) does not have an equal number of rows and columns, it is physically impossible for its RREF to be "the" Identity matrix. The dimensions do not match.

The "Identity Submatrix" in Underdetermined Matrices (e.g., 3x4)

However, non-square matrices can contain the Identity matrix *inside* of them after reduction. Consider a 3x4 matrix A with full row rank. When we reduce it to RREF, we can have at most 3 pivots. If the first 3 columns are linearly independent, the RREF will look like this:

Notice that the first 3 columns of this RREF matrix form a perfect 3 × 3 Identity matrix. The fourth column represents the coefficients of the free variable or the augmented constants. This is often written in block form as [ I₃ | v ].

RREF of Overdetermined Matrices (e.g., 4x3)

For an overdetermined matrix (more rows than columns, such as 4x3) with full column rank, we can have at most 3 pivots. The RREF will look like this:

In this case, the top 3 rows form the 3x3 Identity matrix, while the bottom row consists entirely of zeros (reflecting that the fourth row was a linear combination of the others). So while the full 4x3 matrix is not the Identity, it contains the Identity submatrix at the top.