Rank of a Matrix
The concept of a matrix's rank is a central pillar of linear algebra, which we'll dive into throughout this lesson.
The rank of a matrix represents the maximum number of linearly independent rows or columns that can be found within that matrix.
This piece of information becomes crucial when it comes to solving linear systems or examining linear applications.
$$ rank(A) $$
We often express the rank of a matrix in different ways, such as rank(A), rg(A), r(A), ρ(A). In English, you might also see rank(A) or rk(A).
Some also refer to the rank as the matrix's characteristic.
So, how do you calculate the rank? Determining the rank of a matrix varies depending on the matrix's size. If you're dealing with a small square or rectangular matrix, the criterion of minors method is often utilized. This approach entails finding the highest order (n) of square sub-matrices with non-zero determinants within the larger matrix. For larger matrices, other methods like the Gauss algorithm, which transforms the matrix into a row-echelon form without altering the rank, are often more suitable.
Let's walk through a practical example.
Suppose we have matrix M, comprised of 2 rows and 3 columns.
$$ M = \begin{pmatrix} 3 & 1 & 0 \\ 2 & 1 & 3 \end{pmatrix} $$
Given that M is relatively small, applying the criterion of minors to find the rank would be the logical approach.
Three square matrices of order n=2, meaning with two rows and two columns, can be identified within matrix M:
$$ \begin{pmatrix} 3 & 1 \\ 2 & 1 \end{pmatrix} $$
$$ \begin{pmatrix} 3 & 0 \\ 2 & 3 \end{pmatrix} $$
$$ \begin{pmatrix} 1 & 0 \\ 1 & 3 \end{pmatrix} $$
Next, we calculate the determinant of these square sub-matrices.
$$ \Delta \begin{pmatrix} 3 & 1 \\ 2 & 1 \end{pmatrix} = 3 \cdot 1 - 1 \cdot 2 = 3-2 =1 $$
$$ \Delta \begin{pmatrix} 3 & 0 \\ 2 & 3 \end{pmatrix} = 3 \cdot 3 - 0 \cdot 2 = 9 $$
$$ \Delta \begin{pmatrix} 1 & 0 \\ 1 & 3 \end{pmatrix} = 1 \cdot 3 - 0 \cdot 1 = 3 $$
In our example, at least one matrix presents a non-zero determinant. Hence, the rank of matrix M is r=n=2.
$$ r(M) = 2 $$
Do note that in actuality, you can cease your calculations as soon as you find a sub-matrix with a non-zero determinant, as this indicates the matrix's rank.
Now, what if all the square sub-matrices show a null determinant?
Well, in that case, you would simply repeat the process with sub-matrices of a lower order.
Moving forward, if you appreciated the clarity of this linear algebra lesson, do continue to tune in to our discussions.
The Rank of a Matrix: Key Properties
The rank of a matrix isn't merely an arbitrary number. It holds various critical properties:
- The rank is zero only for the null matrix.
- The rank of a matrix is equal to the rank of its transpose.
- The rank is always less than or equal to both the number of rows and columns.
- If a matrix is invertible, then it has maximum rank.
- The rank of a matrix is a complete invariant for right-left equivalent matrices.
- The rank equates to the dimension of a subspace generated by the matrix's rows or columns.
- The rank represents the dimension of the image of the linear application associated with the matrix.
- The rank stands as the highest order of an invertible minor within the matrix.
What are the applications of the rank?
The rank plays a pivotal role in various applications of linear algebra, including but not limited to solving systems of linear equations, examining linear transformations, and analyzing matrix properties.
Its importance extends to matrix theory, profoundly affecting the structural understanding and properties of matrices.
