Rank of Matrices | Control Systems - Electrical Engineering (EE) PDF Download

What is the Rank of a Matrix?

The rank of a matrix is the order of the highest ordered non-zero minor. Let us consider a non-zero matrix A. A real number 'r' is said to be the rank of the matrix A if it satisfies the following conditions:

• every minor of order r + 1 is zero.
• There exist at least one minor of order 'r' that is non-zero.

The rank of a matrix A is denoted by ρ (A). Here, "ρ" is a Greek letter that should be read as "rho". So ρ (A) should be read as "rho of A" (or) "rank of A".

How to Find the Rank of a Matrix?

The rank of a matrix can be found using three methods. The most easiest of these methods is "converting matrix into echelon form".

• Minor method
• Using echelon form
• Using normal form

Let us study each of these methods in detail.

Finding Rank of a Matrix by Minor Method

Here are the steps to find the rank of a matrix A by the minor method.

• Find the determinant of A (if A is a square matrix). If det (A) ≠ 0, then the rank of A = order of A.
• If either det A = 0 (in case of a square matrix) or A is a rectangular matrix, then see whether there exists any minor of maximum possible order is non-zero. If there exists such non-zero minor, then rank of A = order of that particular minor.
• Repeat the above step if all the minors of the order considered in the above step are zeros and then try to find a non-zero minor of order that is 1 less than the order from the above step.

Here is an example.

Example: Find the rank of the matrix ρ (A) if
Solution: A is a square matrix and so we can find its determinant.
det (A) = 1 (45 - 48) - 2 (36 - 42) + 3 (32 - 35)
= -3 + 12 - 9
= 0
So ρ (A) ≠ order of the matrix. i.e., ρ (A) ≠ 3.
Now, we will see whether we can find any non-zero minor of order 2.

So there exists a minor of order 2 (or 2 × 2) which is non-zero. So the rank of A, ρ (A) = 2.

Rank of a Matrix Using Echelon Form

In the above example, what if the first minor of order 2 × 2 that we found was zero? We had to find all possible minors of order 2 × 2 until we get a non-zero minor to make sure that the rank is 2. This process may be tedious if the order of the matrix is a bigger number. To make the process of finding the rank of a matrix easier, we can convert it into Echelon form. A matrix 'A' is said to be in Echelon form if it is either in upper triangular form or in lower triangular form. We can use elementary row/column transformations and convert the matrix into Echelon form.

A row (or column) transformation can be one of the following:

• Interchanging two rows.
• Multiplying a row by a scalar.
• Multiplying a row by a scalar and then adding it to the other row.

Here are the steps to find the rank of a matrix.

• Convert the matrix into Echelon form using row/column transformations.
• Then the rank of the matrix is equal to the number of non-zero rows in the resultant matrix.

A non-zero row of a matrix is a row in which at least one element is non-zero.

Example: Find the rank of the matrix (the same matrix as in the previous example) by converting it into Echelon form.
Solution:

Now it is in Echelon form and so now we have to count the number of non-zero rows.

The number of non-zero rows = 2 = rank of A.

Therefore, ρ (A) = 2.

Note that we had got the same answer when we calculated the rank using minors.

Rank of a Matrix Using Normal Form

If a rectangular matrix A can be converted into the form by using the elementary row transformations, then A is said to be in normal form. Here, I_r is the identity matrix of order "r" and when A is converted into the normal form, its rank is, ρ (A) = r. Here is an example. Converting into normal form is helpful in determining the rank of a rectangular matrix. But it can be used to find the rank of square matrices also

Column Rank and Row Rank of a Matrix

When we have calculated the rank of the matrix using echelon form and normal form, we have seen that the rank of the matrix is equal to the number of non-zero rows in the reduced form of matrix. This is actually known as "row rank of matrix" as we are counting the number of non-zero "rows". Similarly, the column rank is the number of non-zero columns, or in other words, it is the number of linearly independent columns. For example, in the above example (of the previous section),

• Row rank = the number of non-zero rows = 3
• Column rank = the number of non-zero columns = 3

It is very clear from this that "row rank = column rank" here. This is in fact true for any matrix.

Properties of Rank of a Matrix

• If A is a nonsingular matrix of order n, then its rank is n. i.e., ρ (A) = n.
• If A is in Echelon form, then the rank of A = the number of non-zero rows of A.
• If A is in normal form, then the rank of A = the order of the identity matrix in it.
• If A is a singular matrix of order n, then ρ (A) < n.
• If A is a rectangular matrix of order m x n, then ρ (A) ≤ minimum {m, n}.
• The rank of an identity matrix of order n is n itself.
• The rank of a zero matrix is 0.

Important Notes on Rank of a Matrix:

• While converting the matrix into echelon form or normal form, we can either use row or column transformations. We can also use a mix of row and column transformations.
• To find the rank of a matrix by converting it into echelon form or normal form, we can either count the number of non-zero rows or non-zero columns.
• Column rank = row rank for any matrix.
• The rank of a square matrix of order n is always less than or equal to n.