gen-define-the-rank-of-a-matrix-and-give-one-short-worked-exampl-35e9

Define the rank of a matrix and give one short worked example.

# Rank of a Matrix **Definition:** The **rank** of a matrix is the maximum number of linearly independent rows (or columns) in the matrix. Equivalently, it's the dimension of the vector space spanned by its rows or columns. ## Worked Example Find the rank of matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 2 \end{pmatrix}$ **Solution:** Use row reduction to echelon form: $$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 2 \end{pmatrix}$$ $R_2 - 2R_1 \rightarrow R_2$ and $R_3 - R_1 \rightarrow R_3$: $$\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & -1 & -1 \end{pmatrix}$$ Swap $R_2$ and $R_3$: $$\begin{pmatrix} 1 & 2 & 3 \\ 0 & -1 & -1 \\ 0 & 0 & 0 \end{pmatrix}$$ This echelon form has **2 non-zero rows**. Therefore, **rank(A) = 2**