The rank of a matrix is the maximum number of its linearly independent columns (or rows). The rank of a matrix cannot be greater than the number of its rows or columns.

The rank of a square matrix, $\rho(A)$, is equal to its order, $m$, if it is nonsingular. This means that its columns (rows) are linearly independent.

The rank of a null matrix is zero. A null matrix has no non-zero rows or columns. Therefore, there are no independent rows or columns. Consequently, the rank of a null matrix is zero.

How to find the Rank of a Matrix?

To find the rank of a matrix, we will transform that matrix into its echelon form.

Determine the rank by the number of non-zero rows.

Consider the following matrix:

(\begin{array}{l}A= \begin{bmatrix} 2 & 4 &6 \ 4& 8& 12 \end{bmatrix}\end{array})

We can observe that the second row is twice the size of the first row, however, this does not affect the rank, which is still considered to be 1.

Consider the unit matrix

(\begin{array}{l}A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \end{array})

The rank of this matrix is 3, since the rows are independent.

The rank of a unit matrix of order m is m.

If A matrix is of order m×n, then the rank of A (ρ(A)) is less than or equal to the minimum of m and n (min{m, n } = minimum of m, n).

The rank of A = n, if A is of order n×n and |A| ≠ 0.

If A is of order $n \times n$ and $|A| = 0$, then the rank of A will be less than $n$.

Rank of a Matrix by Row: Echelon Form

We can transform a given non-zero matrix to a simplified form called a Row-echelon form, using the row elementary operations. In this form, we may have rows all of whose entries are zero. Such rows are called zero rows. A non-zero row is one in which at least one of the elements is not zero.

For example, consider the following matrix:

(\begin{array}{l}A = \begin{bmatrix} 1 & 0 & 2 \ 0 & 0 & 1 \ 0 & 0 & 0 \end{bmatrix} \end{array})

Here R1 and R2 are both non-zero rows.

R3 has 0 rows.

A matrix A is said to be in a row-echelon form if:

1. All non-zero rows are above any rows of all zeros
2. The leading coefficient (i.e. the leftmost non-zero entry) of a non-zero row is always strictly to the right of the leading coefficient of the row above it.

(i) Every non-zero row of A occurs above all zero rows of A.

(ii) The first non-zero element in any row i of A occurs in the _j_th column of A, and then all other elements in the _j_th column of A below the first non-zero element of row i are zeros.

(iii) The first non-zero entry in the _i_th row of A lies to the left of the first non-zero entry in the _(i+1)_th row of A.

Note: A matrix is said to be in row-echelon form if all zero rows are located at the bottom of the matrix and if the first non-zero element in any lower row is located to the right of the first non-zero element in the higher row.

If a matrix is in row-echelon form, then all elements below the leading diagonal are equal to 0.

Consider the following matrix:

(\begin{array}{l}A = \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 5 \ 0 & 0 & 0 \end{bmatrix} \end{array})

Matrix A is in row echelon form because when starting from the last row of the matrix, the third row is a zero row, and the first non-zero element in the second row occurs in the third column, which lies to the right of the first non-zero element in the first row, which occurs in the second column.

Also Read

Types of Matrices

Matrix Operations

Adjoint and Inverse of a Matrix

Rank of a Matrix: Solved Examples

Example 1:

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This sentence has been rewritten.

Find the rank of matrix A using the row echelon form.

(\begin{array}{l}A = \begin{bmatrix} 1 & 2 & 3 \ 2 & 1 & 4 \ 3 & 0 & 5 \end{bmatrix} \end{array})

Given:

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Solution:

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Given, $$A = \begin{bmatrix} 1 &2 &3 \ 2& 1 & 4\ 3 & 0 & 5 \end{bmatrix}$$

Now we apply elementary transformations.

R2 - 2R1 → R2

R3 - 3R1

We get

(\begin{bmatrix} 1 &2 &3 \ 0& -3 & -2\ 0 & -6 & -4 \end{bmatrix})

R3 - 2R2 → 0

(\begin{bmatrix} 1 &2 &3 \ 0& -3 & -2\ 0 & 0 & 0 \end{bmatrix})

The matrix is in row echelon form.

Number of non-zero rows: 2

The rank of matrix A is 2.

Example 2:

The rank of matrix A is 2.

Given:

Welcome to our store

Solution:

Welcome to Our Store

Given, $$A = \begin{bmatrix} 1 &2 &3 \ 2& 3 &4\ 3 & 5 & 7 \end{bmatrix}$$

Now we transform the matrix A to echelon form by using elementary transformations.

R2 - 2R1 → R2

R3 - 3R1

(\begin{bmatrix} 1 &2 &3 \ 0& -1 &-2\ 0 & -1 & -2 \end{bmatrix})

R3 - (R3 - R2)

(\begin{bmatrix} 1 &2 &3 \ 0& -1 &-2\ 0 & 0 & 0 \end{bmatrix})

Number of non-zero rows: 2

The rank of matrix A is 2.

Example 3:

What is the rank of the matrix?

(\begin{array}{l}\begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1 \end{bmatrix}\end{array})

Given:

Hello World

Solution:

Hello World

Given

This is a given statement.

Given This is a given statement.

(\begin{bmatrix} 1 &1 &1 \ 1& 1 &1\ 1 & 1 & 1 \end{bmatrix})

R2 - R1 → R2

R3 - (R3 - R1)

We get

(\begin{bmatrix} 1 & 1 & 1 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix})

The number of non-zero rows here is 1.

The rank of the matrix is 1.

Example 4:

The rank of the 2×2 matrix $\begin{array}{l}B = \begin{bmatrix} 5 & 6\ 7& 8 \end{bmatrix}\end{array}$ is 2.

Given:

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Solution:

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Given, $$B = \begin{bmatrix} 5 & 6\ 7 & 8 \end{bmatrix}$$

Order of B = 2 x 2

|B| = -2 ≠ 0

The rank of B is 2.

Example 5:

Given, $$A = \begin{bmatrix} 4& 7\ 8& 14 \end{bmatrix}$$

What is the size of matrix A?

Given:

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Solution:

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Given

(\begin{array}{l}A = \begin{bmatrix} 4 & 8 \ 7 & 14 \end{bmatrix}\end{array} )

By observing the rows, we can see that the elements of the second row are twice as large as the elements of the first row.

2R1 - R2

(\begin{array}{l}\begin{bmatrix} 0 & 0 \ 8 & 14 \end{bmatrix}\end{array})

Number of non-zero rows: 1

The rank of matrix A is 1.

Example 6:

Example 6:

The rank of matrix M is

(\begin{array}{l}M = \begin{bmatrix} 0 & 1 & 1\ 1& 0 &1 \ 1& 1& 0 \end{bmatrix}\end{array})

a) 1

b) 2

c) 3

d) 0

Given:

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Solution:

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(\begin{array}{l}M = \begin{bmatrix} 0 & 1 & 1\ 1& 0 &1 \ 1& 1& 0 \end{bmatrix}\end{array} )

Apply row transformations to put the matrix into echelon form.

Swap R1 and R2.

(\begin{bmatrix} 1 & 0 & 1 \ 0 & 1 & 1 \ 1 & 1 & 0 \end{bmatrix})

R3 - (R3 - R1)

(\begin{bmatrix} 1 & 0 & 1 \ 0 & 1 & 1 \ 0 & 1 & -1 \end{bmatrix})

R3 - R2 → R3

(\begin{array}{l}\begin{bmatrix} 1 & 0 & 1\ 0& 1 &1 \ 0& 0& -2 \end{bmatrix}\end{array} )

R3/-2

(\begin{bmatrix} 1 & 0 & 1 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{bmatrix})

Rank = 3, since there are three non-zero rows.

Therefore, option (c) is the answer.

Example 7: This example demonstrates how to rewrite a statement.

This example shows how to rephrase a statement.

The rank of the following matrix is

a) 1

b) 2

c) 3

4. 4

Given:

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Solution:

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Given, $$\begin{bmatrix} 1 & 1 & 0& -2\ 2& 0& 2 & 2\ 4& 1 & 3 & 1 \end{bmatrix}$$

We transform the matrix using elementary row operations.

R2 - 2R1 → R2

(\begin{bmatrix} 1 & 1 & 0& -2\ 0& -2& 2 & 6\ 0& -3 & 3 & 9 \end{bmatrix})

R2/-2 → R2

(\begin{array}{l}\begin{bmatrix} 1 & 1 & 0 & -2 \ 0 & 1 & -1 & -3 \ 0 & -3 & 3 & 9 \end{bmatrix}\end{array})

R3 + 3R2 → R3

(\begin{array}{l}\begin{bmatrix} 1 & 1 & 0 & -2 \ 0 & 1 & -1 & -3 \ 0 & 0 & 0 & 0 \end{bmatrix}\end{array})

The rank is 2 since there are 2 non-zero rows.

Therefore, answer (b) is correct.

Example 8: This is an example.

This is Example 8: This is an example.

The rank of (P + Q) = 3

a) 1

b) 0

c) 2

d) 3

Given:

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Solution:

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Given, $$P = \begin{bmatrix} 1 & 1 & -1\ 2 & -3& 4\ 3 & -2 & 3 \end{bmatrix}$$

(\begin{array}{l}Q = \begin{bmatrix} -1 & -2 & -1 \ 6 & 12 & 6 \ 5 & 10 & 5 \end{bmatrix} \end{array})

(\begin{array}{l}Q + P = \begin{bmatrix} 0 & -1 & -2\ 8& 9& 10\ 8& 8 & 8 \end{bmatrix}\end{array} )

Swap C1 and C2

(\begin{array}{l}\begin{bmatrix} -1 & 0 & -2 \ 9 & 8 & 10 \ 8 & 8 & 8 \end{bmatrix}\end{array})

R2 + 9R1 → R2

R3 + 8R1 → R3

(\begin{bmatrix} -1 & 0 & -2\ 0& 8& -8\ 0& 8 & -8 \end{bmatrix})

R3 - R2 → R3

(\begin{array}{l}\begin{bmatrix} -1 & 0 & -2 \ 0 & 8 & -8 \ 0 & 0 & 0 \end{bmatrix}\end{array})

R2/8 → R2

(\begin{array}{l}\begin{bmatrix} -1 & 0 & -2 \ 0 & 1 & -1 \ 0 & 0 & 0 \end{bmatrix}\end{array})

Number of non-zero rows: 2

So the rank is 2

Therefore, option (c) is the correct answer.

Video Lessons

Matrices and Determinants - Important Topics

Matrices and Determinants - Important Questions

Frequently Asked Questions

What is the rank of a matrix?

The rank of a matrix is the maximum number of its linearly independent rows (or columns).

Does the rank of a matrix exceed the number of rows or columns?

The rank of a matrix cannot be greater than the number of its rows or columns.

What are Rank Zero Matrices?

If all elements of a matrix become zero, then the matrix is a rank zero matrix.

What is the Rank of a Matrix?

To find the rank of a matrix, transform the matrix into its echelon form. Then count the number of non-zero rows to determine the rank.