3x3 Matrix Rank

What is Matrix Rank?

The rank of a matrix is a fundamental concept in linear algebra. It represents the dimension of the vector space spanned by the columns (or rows) of the matrix. In other words, it's the number of linearly independent columns or rows in the matrix.

The Matrix Rank Formula

For a 3x3 matrix A, the rank can be determined by the following steps:

1. Calculate the determinant of A. If it's non-zero, rank(A) = 3.
2. If det(A) = 0, check all 2x2 submatrices. If any have a non-zero determinant, rank(A) = 2.
3. If all 2x2 determinants are zero, check if any element is non-zero. If so, rank(A) = 1.
4. If all elements are zero, rank(A) = 0.

Calculation Steps

To calculate the rank of a 3x3 matrix:

1. Compute the determinant of the full matrix: \[\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]
2. If det(A) ≠ 0, rank(A) = 3
3. If det(A) = 0, check all 2x2 submatrices: \[\det\begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}, \det\begin{pmatrix}a_{11} & a_{13} \\ a_{21} & a_{23}\end{pmatrix}, \text{ etc.}\]
4. If any 2x2 determinant ≠ 0, rank(A) = 2
5. If all 2x2 determinants = 0, check if any element ≠ 0
6. If any element ≠ 0, rank(A) = 1; otherwise, rank(A) = 0

Example Calculation

Let's calculate the rank of matrix A:

\[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\]

1. Calculate det(A):

\[\begin{aligned} \det(A) &= 1(5\cdot9 - 6\cdot8) - 2(4\cdot9 - 6\cdot7) + 3(4\cdot8 - 5\cdot7) \\ &= 1(-3) - 2(-6) + 3(-3) \\ &= -3 + 12 - 9 \\ &= 0 \end{aligned}\]

2. Check 2x2 submatrices:

\[\det\begin{pmatrix}1 & 2 \\ 4 & 5\end{pmatrix} = 1\cdot5 - 2\cdot4 = -3 \neq 0\]

Since we found a non-zero 2x2 determinant, the rank of A is 2.

Visual Representation

This diagram visually represents the 3x3 matrix A with rank 2.