The rank of a matrix is the number of lines in the matrix (rows or columns) that are linearly independent.

A line is linearly dependent on another one or others when a linear combination between them can be established.

A line is linearly independent of another one or others when a linear combination between them cannot be established.

The rank of a matrix is symbolized as rank(A) or r(A).

Calculating the Rank of a Matrix

The Gaussian elimination method is used to calculate the rank of a matrix.

A line can be discarded if:

  1. All the coefficients are zeros.
  2. There are two equal lines.
  3. A line is proportional to another.
  4. A line is a linear combination of others.

\begin{pmatrix} 1 & 2 & -1 & 3 & -2 \\ 2 & 1 & 0 & 1 & 1 \\ 2 & 4 & -2 & 6 & -4 \\ 0 & 0 & 0 & 0 & 0 \\ 5 & 4 & -1 & 5 & 0 \end{pmatrix}

{ r }_{ 3 } = 2 . { r }_{ 1 } \rightarrow discarded because of point 3

{ r }_{ 4 } \rightarrow discarded because of point 1

{ r }_{ 5 }  = 2 { r }_{ 2 }  + { r }_{ 1 } \rightarrow discarded because of point 4

Since only two rows are unique that means the rank of matrix is r(A) = 2

 

In general, eliminate the maximum possible number of lines, and the range is the number of nonzero rows.

A = \begin{pmatrix} 1 & -4 & 2 & -1 \\ 3 & -12 & 6 & -3 \\ 2 & -1 & 0 & 1 \\ 0 & 1 & 3 & -1 \end{pmatrix}

{ r }_{ 2 } = { r }_{ 2 } - 3 { r }_{ 1 }

{ r }_{ 3 } = { r }_{ 3 } - 2 { r }_{ 1 }

Therefore r(A) = 3

 

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Example

Calculate the rank of the following matrix:

\begin{pmatrix} 2 & -1 & 0 & 7 \\ 1 & 0 & 1 & 3 \\ 3 & 2 & 7 & 7 \end{pmatrix}

{ r }_{ 1 } = { r }_{ 1 } - 2 { r }_{ 2 }

\begin{pmatrix} 0 & -1 & -2 & 1 \\ 1 & 0 & 1 & 3 \\ 3 & 2 & 7 & 7 \end{pmatrix}

{ r }_{ 3 } = { r }_{ 3 } - 3 { r }_{ 2 }

\begin{pmatrix} 0 & -1 & -2 & 1 \\ 1 & 0 & 1 & 3 \\ 0 & 2 & 4 & -2 \end{pmatrix}

{ r }_{ 3 } = { r }_{ 3 } + 2 { r }_{ 1 }

\begin{pmatrix} 0 & -1 & -2 & 1 \\ 1 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{pmatrix}

Therefore, r(A) = 2.

 

Calculating the rank of a matrix for determimants

 

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.