Can a matrix be injective?

In the realm of mathematics, matrices are fundamental structures that encapsulate numerical data in a rectangular array. When discussing properties of matrices, a key concept is the row reduced form, denoted as Ared, of a given matrix A. This transformed matrix serves as a cornerstone in analyzing various attributes of A.

One such attribute is injectivity, a term borrowed from set theory to describe functions that map distinct elements of the domain to distinct elements of the codomain. In the context of matrices, injectivity refers to the ability of A to preserve the distinctness of its column vectors.

To determine if a matrix A is injective, we turn to its row reduced form Ared. This process simplifies A by applying a series of row operations, transforming it into a form that reveals its essential structural properties.

If, upon inspection of Ared, we find that every column contains a leading 1 (the first non-zero element in each column), this indicates that A is injective. The presence of leading 1s in every column signifies that no column of A can be expressed as a linear combination of the others, thus preserving the distinctiveness of its column vectors.

Conversely, if Ared has at least one column without a leading 1, it implies that the corresponding column in A can be linearly represented by the other columns. This lack of independence among the columns of A undermines its ability to be injective, as distinct input vectors may map to the same output vector.