Rank of 3×3 Matrix Calculator
This calculator determines the rank of a 3×3 matrix using row reduction (Gaussian elimination) or determinant methods. It is useful for students, engineers, and professionals working with linear algebra and solving matrix equations.
3×3 Matrix Rank Solver
Rank of 3×3 Matrix Formula
Where:
- $$A$$ is a 3×3 matrix
- $$\det(A)$$ is the determinant of $$A$$
- Minors are determinants of 2×2 submatrices
Rank of 3×3 Matrix – Calculation Example
$$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$
$$\det(A) = 1(5 \cdot 9 – 6 \cdot 8) – 2(4 \cdot 9 – 6 \cdot 7) + 3(4 \cdot 8 – 5 \cdot 7) = 1(-3) – 2(-6) + 3(-3) = -3 + 12 – 9 = 0$$
Since $$\(\det(A)=0\)$, check 2×2 minors:
$$\det \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix} = 1 \cdot 5 – 2 \cdot 4 = 5 – 8 = -3 \neq 0 $$
So, rank(A) = 2.
The rank of a 3×3 matrix shows the number of linearly independent rows or columns. It is widely used in solving linear systems, determining matrix invertibility, and understanding matrix properties. Our calculator automates the process, providing accurate rank determination for any 3×3 matrix.