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.
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