ASL STEM

Definition

The Fundamental Theorem of Matrices is as follows: Let A be an n X n matrix. The following statements are equivalent: (a) A is invertible. (b) Ax = b has a unique solution for every b in R^n. (c) Ax = 0 has only the trivial solution. (d) The reduced row echelon form of A is I_n. (e) A is a product of elementary matrices. (f) rank(A) = n. (g) nullity(A) = 0. (h) The column vectors of A are linearly independent. (i) The column vectors of A span R^n. (j) The column vectors of A form a basis for R^n. (k) The row vectors of A are linearly independent. (l) The row vectors of A span R^n. (m) The row vectors of A form a basis for R^n. (n) det A does not equal 0. (o) 0 is not an eigenvalue of A.

Source: Linear Algebra: A Modern Introduction, 3rd edition by David Poole (note-custom edition titled Matrix Algebra)