## Quick Reference

(of a matrix)

Let **A** be an *m*×*n* matrix. The column rank of **A** is the largest number of elements in a linearly independent set of columns of **A**. The row rank of **A** is the largest number of elements in a linearly independent set of rows of **A**. It can be shown that elementary row operations on a matrix do not change the column rank or the row rank. Consequently, the column rank and row rank of **A** are equal, both being equal to the number of non-zero rows in the matrix in reduced echelon form to which **A** can be transformed. This common value is the rank of **A**. It can also be shown that the rank of **A** is equal to the number of rows and columns in the largest square submatrix of **A** that has non-zero determinant. An *n*×*n* matrix is invertible if and only if it has rank *n*.

*Subjects:*
Mathematics.