Lecture 4 — Rank of a Matrix: Definition, Properties, 10 Examples (3×3 & 4×4) + Calculator

Lecture 4: Rank of a Matrix

1. Definition

The rank of a matrix A, denoted rank(A), is the maximum number of linearly independent rows (row rank) or columns (column rank) of the matrix. Equivalently, it is the number of pivot (leading 1) rows in any row-echelon form or reduced row-echelon form (RREF) of the matrix.

Formally, for an m × n matrix, 0 ≤ rank(A) ≤ min(m, n).

2. Important Properties of Rank

  1. Bounds: 0 ≤ rank(A) ≤ min(m,n).
  2. Zero matrix: rank(0) = 0.
  3. Invertible matrices: If A is an invertible n × n matrix then rank(A) = n.
  4. Row operations: Elementary row operations do not change the row space dimension (rank).
  5. Transpose: rank(A) = rank(Aᵀ).
  6. Rank inequalities: rank(A+B) ≤ rank(A) + rank(B).
  7. Multiplication: rank(AB) ≤ min(rank(A), rank(B)).
  8. Column/Row equivalence: The maximum number of independent columns equals the maximum number of independent rows.

3. Examples

3×3 matrices

  1. Matrix 1: \(A_1=\begin{bmatrix}1 & 2 & 3\\ 2 & 4 & 6\\ 3 & 6 & 9\end{bmatrix}\)
    Explanation: Every row is a multiple of the first row (Row2 = 2·Row1, Row3 = 3·Row1). So only one independent row. rank = 1
  2. Matrix 2: \(A_2=\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}\)
    Explanation: Identity matrix — all three rows are independent. rank = 3
  3. Matrix 3: \(A_3=\begin{bmatrix}1 & 2 & 3\\ 1 & 2 & 3\\ 0 & 1 & 4\end{bmatrix}\)
    Explanation: Row2 = Row1, so drop one duplicate; remaining rows Row1 and Row3 are independent. rank = 2
  4. Matrix 4: \(A_4=\begin{bmatrix}2 & 4 & 6\\ 1 & 2 & 3\\ 0 & 0 & 0\end{bmatrix}\)
    Explanation: Row1 = 2·Row2, Row3 is zero — only one independent row. rank = 1
  5. Matrix 5: \(A_5=\begin{bmatrix}1 & 2 & 0\\ 0 & 1 & 3\\ 1 & 3 & 3\end{bmatrix}\)
    Explanation: Subtract R1 from R3 → (0,1,3) = R2; so Row3 = R1 + R2 ⇒ only two independent rows. rank = 2
  6. Matrix 6: \(A_6=\begin{bmatrix}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 10\end{bmatrix}\)
    Explanation: Determinant ≠ 0 (compute to check) ⇒ full rank. rank = 3

4×4 matrices

  1. Matrix 7: \(B_1=\begin{bmatrix}1 & 0 & 2 & 1\\ 0 & 1 & 3 & 2\\ 1 & 1 & 5 & 3\\ 2 & 1 & 7 & 4\end{bmatrix}\)
    Explanation: Rows 3 and 4 are linear combinations of rows 1 and 2 (elimination shows two pivots). rank = 2
  2. Matrix 8: \(B_2=\begin{bmatrix}1 & 2 & 3 & 4\\ 2 & 4 & 6 & 8\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\end{bmatrix}\)
    Explanation: Row2 = 2·Row1, drop it; remaining rows 1,3,4 are independent ⇒ rank = 3. rank = 3
  3. Matrix 9: \(B_3=\begin{bmatrix}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{bmatrix}\)
    Explanation: 4×4 identity matrix — full rank. rank = 4
  4. Matrix 10: \(B_4=\begin{bmatrix}1 & 2 & 3 & 4\\ 2 & 4 & 6 & 8\\ 3 & 6 & 9 & 12\\ 4 & 8 & 12 & 16\end{bmatrix}\)
    Explanation: All rows are multiples of Row1 ⇒ only one independent row. rank = 1

4. Interactive Rank Calculator (with RREF steps)

Enter a matrix as rows separated by semicolons and numbers separated by commas or spaces. Examples: 1,2,3;4,5,6;7,8,9 or 1 2 3 4; 2 4 6 8; 0 1 0 1; 1 0 1 0.

Rank: -
Steps will appear here (rendered with MathJax)

Video