Rank of a Matrix

The rank of a matrix is equal to the number of linearly independent rows (or columns) in it. Hence, it cannot more than its number of rows and columns. For example, if we consider the identity matrix of order 3 × 3, all its rows (or columns) are linearly independent and hence its rank is 3.

Let us learn more about the rank of a matrix along with its mathematical definition and let us see how to find the rank of the matrix along with examples.

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".

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.

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 A = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]\).

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.

\(\left|\begin{array}{ll}
1 & 2 \\ \\
4 & 5
\end{array}\right|\) = 5 - 8 = -3 ≠ 0.

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

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 A = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]\) (the same matrix as in the previous example) by converting it into Echelon form.

Solution:

Given matrix is, A = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]\).

Apply R₂ → R₂ - 4R₁ and R₃ → R₃ - 7R₁, we get:

\(\left[\begin{array}{lll}
1 & 2 & 3 \\
0 & -3 & -6 \\
0 & -6 & -12
\end{array}\right]\)

Now, we apply R₃ → R₃ - 2R₂, we get:

\(\left[\begin{array}{lll}
1 & 2 & 3 \\
0 & -3 & -6 \\
0 & 0 & 0
\end{array}\right]\)

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.

If a rectangular matrix A can be converted into the form \(\left[\begin{array}{ll}
I_r & 0 \\ \\
0 & 0
\end{array}\right]\) 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.

Example: Find the rank of the matrix A = \(\left[\begin{array}{lll}
1 & 2 & 1&2 \\
1 & 3 & 2 & 2 \\
2 & 4 & 3 & 4 \\
3 & 7 & 4 & 6
\end{array}\right]\) (again the same matrix) by converting it into normal form.

Solution:

Apply R₂ → R₂ - R₁, R₃ → R₃ - 2R₁, and R₄ → R₄ - 3R₁ we get:

\(\left[\begin{array}{lll}
1 & 2 & 1&2 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 1 & 0
\end{array}\right]\)

Now apply, R₁ → R₁ - 2R₂ and R₄ → R₄ - R₂,

\(\left[\begin{array}{lll}
1 & 0 & -1&2 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]\)

Apply R₁ → R₁ + R₃ and R₂ → R₂ - R₃,

\(\left[\begin{array}{lll}
1 & 0 & 0 &2 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]\)

Now apply C₄ → C₄ - 2C₁,

\(\left[\begin{array}{lll}
1 & 0 & 0 &0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]\)

This is same as \(\left[\begin{array}{ll}
I_3 & 0 \\ \\
0 & 0
\end{array}\right]\).

Therefore, the rank of A is, ρ (A) = 3.

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.

• 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.

☛ Related Topics:

• Determinant Calculator
• Eigenvalue Calculator
• Matrix Addition Calculator
• Matrix Multiplication Calculator

What is the Definition of Rank of a Matrix?

The rank of a matrix is the number of linearly independent rows or columns in it. The rank of a matrix A is denoted by ρ (A) which is read as "rho of A". For example, the rank of a zero matrix is 0 as there are no linearly independent rows in it.

How to Find the Rank of the Matrix?

To find the rank of a matrix, we can use one of the following methods:

• Find the highest ordered non-zero minor and its order would give the rank.
• Convert the matrix into echelon form using the row/column operations. Then the number of non-zero rows in it would give the rank of the matrix.
• Convert the matrix into the normal form \(\left[\begin{array}{ll}
  I_r & 0 \\ \\
  0 & 0
  \end{array}\right]\), where I_r is the identity matrix of order 'r'. Then rank of the matrix = r.

What is the Rank of a Matrix of Order 3 × 3?

The rank of a matrix of order 3 × 3 is 3 if its determinant is NOT 0. If its determinant is 0, then convert it into Echelon form by using row/column transformations, then the number of non-zero rows/columns would give the rank.

What is the Rank of a Matrix of Order 2 × 2?

If the determinant of a 2 × 2 matrix is NOT 0, then its rank is 2. If its determinant is 0, then its rank is either 1 or 0. The exact rank can be found by converting it into echelon form or normal form.

How to Find the Rank of a Matrix Using Determinant?

To find the rank of a matrix of order n, first, compute its determinant (in the case of a square matrix). If it is NOT 0, then its rank = n. If it is 0, then see whether there is any non-zero minor of order n - 1. If such minor exists, then the rank of the matrix = n - 1. If all the minors of order n - 1 are zeros, then we should repeat the process for minors of order n - 2, and so on until we are able to find the rank.

What is the Rank of a Null Matrix?

Null matrix is a square matrix with all entries to be 0s. The determinant of a null matrix and any of its minor is 0 itself. Hence there doesn't exist any minor of a null matrix that is non-zero. Therefore, the rank of a null matrix is 0.

What is the Shortcut to Find the Rank of a Matrix?

If the determinant of a matrix is not zero, then the rank of the matrix is equal to the order of the matrix. This can be used as a shortcut. But this shortcut does not work when the determinant is 0. In this case, we have to use either minors, Echelon form, or normal form to find the rank like how the processes are explained on this page.

What are the Applications of Rank of Matrix?

The rank of a matrix is mainly useful to determine the number of solutions of a system of equations. If a system has 'n' equations in 'n' variables, then, we first find the rank of the augmented matrix and the rank of the coefficient matrix.

• If the rank (augmented matrix) ≠ rank (coefficient matrix), then the system has no solution (inconsistent).
• If the rank (augmented matrix) = rank (coefficient matrix) = number of variables, then the system has a unique solution (consistent).
• If the rank (augmented matrix) = rank (coefficient matrix) < number of variables, then the system has an infinite number of solutions (consistent).

What Does the Rank of a Matrix Tell Us?

The rank of a matrix would give the number of linearly independent rows (or columns). The more the rank of the matrix the more the linearly independent rows and also the more the informative content.

Can the Rank of a Matrix Ever be Greater than the Number of Rows or Columns?

No, the rank of a matrix is always less than or equal to each of the number of rows and number of columns.

What is the Relation Between the Rank of a Matrix and Eigenvalues?

There is a very close relationship between the rank of a matrix and the eigenvalues. The rank of a matrix is exactly equal to the number of non-zero eigenvalues.