Suppose A is an m ×n matrix, and B and C are nonsingular...

Suppose $A$ is an $m \times n$ matrix, and $B$ and $C$ are nonsingular matrices of sizes $m \times m$ and $n \times n$, respectively. Prove that $\operatorname{rank} A=\operatorname{rank} B A=\operatorname{rank} A C=\operatorname{rank} B A C$.

Key Concepts

Rank Preservation under Invertible Transformations

The key concept here is that rank remains invariant when a matrix is multiplied by an invertible matrix. Since left multiplication by an invertible matrix changes the row space and right multiplication changes the column space by mere basis transformations, the intrinsic dimensionality of these spaces, as measured by the rank, stays the same. This principle underpins the proof that rank A equals rank BA, rank AC, and rank BAC.

Change of Basis (Linear Transformation Equivalence)

Multiplying a matrix on the left or right by an invertible matrix corresponds to changing the basis in the domain or codomain of the associated linear transformation. This change of basis does not alter the fundamental properties of the matrix, including its rank, because it only re-expresses the vector space without affecting the linear independence of the vectors.

Matrix Rank

Matrix rank is a measure of the dimension of the vector space generated by the columns (or rows) of the matrix. It indicates the maximum number of linearly independent columns or rows present. Understanding rank is essential for determining the solutions to linear systems and analyzing the behavior of linear transformations.

Nonsingular (Invertible) Matrix

A nonsingular matrix, also known as an invertible matrix, is one that has a two-sided inverse. This property implies that the matrix performs a one-to-one mapping of the space, preserving the linear independence of vectors it acts upon. The invertibility of a matrix guarantees the absence of loss of information when transforming vectors, which is critical in proving rank preservation.

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