Linear Algebra, 5th Edition

3.2 The Rank of a Matrix and Matrix Inverses

In this section, we define the rank of a matrix. We then use elementary operations to compute the rank of a matrix and a linear transformation. The section concludes with a procedure for computing the inverse of an invertible matrix.

Definition.

If $A\in {\text{M}}_{m\times n}(F)$, we define the rank of A, denoted rank(A), to be the rank of the linear transformation ${\text{L}}_A:{\text{F}}^n\to {\text{F}}^m$.

Many results about the rank of a matrix follow immediately from the corresponding facts about a linear transformation. An important result of this type, which follows from Fact 3 (p. 101) and Corollary 2 to Theorem 2.18 (p. 103), is that an $n\times n$ matrix is invertible if and only if its rank is n.

Every matrix A is the matrix representation of the linear ...