# 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. 