Tools

Rank of a Matrix

Definition: The rank of a matrix is the maximum number of linearly independent rows or columns. Rank tells us how many independent equations we actually have.

In practice, we often find the rank by transforming a matrix into Row Echelon Form (REF). Each row in REF has a leading entry called a pivot, which is the first non-zero element in that row. All entries below each pivot are zero. The number of non-zero rows in REF equals the rank of the matrix.

R1, R2, R3 represent the rows of the matrix. When we write operations like R2 ← R2 - R1, it means we are updating row 2 by subtracting row 1 from it. These row operations simplify the matrix to find pivots and calculate the rank.

Example 1: Finding Rank

Input Matrix:

\( A = \begin{bmatrix} 1 & -2 & 1 \\ 1 & -1 & 2 \\ 0 & -2 & 2 \end{bmatrix} \)

Step 1: Eliminate below first pivot (Column 1)

Pivot in column 1 is 1 in R1. Apply row operation to make entries below pivot zero:

  • \( R_2 \leftarrow R_2 - R_1 \)
\( \begin{bmatrix} 1 & -2 & 1 \\ 0 & 1 & 1 \\ 0 & -2 & 2 \end{bmatrix} \)

Step 2: Eliminate below pivot in 2nd column

Pivot in column 2 is 1 in R2. Apply row operation to eliminate entry below pivot:

  • \( R_3 \leftarrow R_3 + 2 R_2 \)
\( \begin{bmatrix} 1 & -2 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 4 \end{bmatrix} \)

Step 3: Final Row Echelon Form (REF)

\( \text{REF}(A) = \begin{bmatrix} 1 & -2 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 4 \end{bmatrix} \)

Count the number of non-zero rows in REF. Here we have 3 non-zero rows, so:

Rank(A) = 3

Example 2: Zero Matrix (No Rank)

\( B = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)

All rows are zero, so there are no pivots.

Rank(B) = 0