Rank of a Matrix

The rank of a matrix is define in several ways. Here we will see most of all the definitions of the rank of a matrix and also we will discuss properties of the rank of a matrix.

Rank of a Matrix

Let $A$ be a non-zero matrix. A positive integer $r\in\mathbb{N}$ is called the rank of the matrix $A$ if there exists a minor of order $r$ having non-zero determinant and all the minors of $A$ of order higher than $r$ are zero if exists. Moreover the rank of a zero matrix is defined to be 0. The rank of the matrix is denoted by $\rho(A)$.

Examples of the rank of a matrix

Let us see some examples see to find the rank of a matrix by above definition.

Rank of a matrix Example 1

Let
$$A=\left[\begin{matrix}1&2&4\\3&6&7\\2&4&8\end{matrix}\right]$$

Here $A$ is a $3\times 3$ matrix. So the highest order minor is of order 3 and it is matrix $A$ it self. Now,
$$|A|=0.$$

Thus we can say all the minors of order 3 have determinant 0. So $\rho(A)\leq 2$. Now the matrix $A$ has 9 minors of order 2. Among them it is easy to see that the minor $\begin{matrix}6&7\\4&8\end{matrix}$ has determinant,

$$ \begin{aligned} \left|\begin{matrix}6&7\\4&8\end{matrix}\right| &=48-28\\ &=20\ne 0. \end{aligned} $$

So the matrix $A$ has a minor of order 2 having non zero determinant and all the minors of order higher than 2 is zero. Hence by definition $\rho(A)=2$.

Rank of a matrix Example 2

Let
$$A=\left[\begin{matrix}1&2&4&3\\2&4&8&6\\4&8&16&12\end{matrix}\right].$$

Here $A$ is a $3\times 4$ matrix. So the highest order minor is of order 3. There are four minors of order 3. Now, the determinant of all these four minors are

$$\left|\begin{matrix}1&2&4\\2&4&8\\4&8&16\end{matrix}\right|=0$$

$$\left|\begin{matrix}1&2&3\\2&4&6\\4&8&12\end{matrix}\right|=0$$

$$\left|\begin{matrix}1&4&3\\2&8&6\\4&16&12\end{matrix}\right|=0$$

$$\left|\begin{matrix}2&4&3\\4&8&6\\8&16&12\end{matrix}\right|=0$$

Thus we can say all the minors of order 3 have determinant 0. So $\rho(A)\leq 2$. Now the matrix $A$ has 18 minors of order 2. By manual varification it is easy to see that all the minors of order 2 have determinant 0. Hence $\rho(A)\leq 1$.

Since $A$ is a non zero matrix, any non-zero entry will serve as a minor having non-zero determinant. Hence $\rho(A)=1$.

Rank of a matrix Example 3

Let
$$A=\left[\begin{matrix}0&0&0\\0&0&0\end{matrix}\right]$$

Since $A$ is a zero matrix by definition $\rho(A)=0$.

If $A$ is a matrix of order $m\times n$, then $\rho(A)\leq min{m,n}$.
Zero matrices are the only matrices having zero rank.

Rank of a Matrix by Row Echelon Form.

As order of a matrix becomes large, the above process of finding rank becomes tedious. But we have some results which will make our job to find the rank of the matrix easy.

Theorem 1.

Let $A$ be a matrix. The rank of the matix $A$ is the number of non-zero rows in the row echelon form of the matrix $A$.

Rank of a matrix Example 4

Find the rank of the matrix
$$A=\left[\begin{matrix}-1&2&3&-2\\2&-5&1&2\\3&-8&5&2\\5&-12&-1&6\end{matrix}\right].$$

Solution

$$A=\left[\begin{matrix}-1&2&3&-2\\2&-5&1&2\\3&-8&5&2\\5&-12&-1&6\end{matrix}\right]$$

Applying $R_{12}(2),\;\;R_{13}(3),\;\;R_{14}(5)$,

$$\sim\left[\begin{matrix}-1&2&3&-2\\0&-1&7&-2\\0&-2&14&-4\\0&-2&14&-4\end{matrix}\right]$$

Applying $R_{23}(-2),\;\;R_{24}(-2)$,

$$\sim\left[\begin{matrix}-1&2&3&-2\\0&-1&7&-2\\0&0&0&0\\0&0&0&0\end{matrix}\right]$$

Which is a row echelon form of the matrix $A$ and it has two non-zero rows. Hence $\rho(A)=2$.

Theorem 2.

Let $A$ be a matrix. The rank of the matix $A$ is the number of linearly independent rows in the matrix $A$. ( Here the rank is also known as row rank of the matrix).

Theorem 3.

Let $A$ be a matrix. The rank of the matix $A$ is the number of linearly independent columns in the matrix $A$. (Here the rank is also known as column rank of the matrix).

Results and Properties of the rank of the Matrix.

  • Let $A$ be a matrix. Then $\rho(A)=\rho(A^T)$.
  • Let $A$ be a non-singular matrix of order $n$. Then $\rho(A)=n$.
  • Let $A$ be a matrix and $B$ be a non singular matrix. Then $\rho(BA)=\rho(A)$ i.e. the multiplication to a matrix by a non-singular matrix does not affect the rank of the matrix.

Endnote

In this tutorial, you learned about Rank of the Matrix. You also learned about how to find the rank of a matrix with illustrated examples.

To learn more about matrix algebra, please refer to the following tutorials:

Matrix Algebra

Let me know in the comments if you have any questions on Rank of the matrix and your thought on this article.

Milan holds Ph.D. Degree in Mathematics, his area of research is Banach Algebra and Harmonic Analysis, a branch of pure mathematics. He has cleared NET and GPSC (Gujarat Public Service Commission) with first rank. Milan enjoys teaching and problem solving in Mathematics. He has good hand on Real Analysis, Complex Analysis, Linear Algebra, Abstract Algebra and Differential Equation. Apart from teaching Milan spends his time to guide the students for various competitive examinations in Mathematics like JAM, CSIR NET-JRF, GSET, GATE, NBHM, TIFR etc. More than 29 students had cleared such type of examinations under his guidance. He has teaching experience of more than 13 years in the field of Mathematics at graduate level in Engineering and Science College.

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