# JEE Main & Advanced Mathematics Determinants & Matrices Rank of Matrix

Rank of Matrix

Category : JEE Main & Advanced

Definition : Let A be a $m\times n$ matrix. If we retain any $r$ rows and $r$ columns of A we shall have a square sub-matrix of order $r$. The determinant of the square sub-matrix of order $r$ is called a minor of A order $r$. Consider any matrix A which is of the order of $3\times 4$ say, $A=\left| \begin{matrix} 1 & 3 & 4 & 5 \\ 1 & 2 & 6 & 7 \\ 1 & 5 & 0 & 1 \\ \end{matrix} \right|$. It is $3\times 4$ matrix so we can have minors of order 3, 2 or 1. Taking any three rows and three columns minor of order three. Hence minor of order $3=\left| \,\begin{matrix} 1 & 3 & 4 \\ 1 & 2 & 6 \\ 1 & 5 & 0 \\ \end{matrix}\, \right|=0$

Making two zeros and expanding above minor is zero. Similarly we can consider any other minor of order 3 and it can be shown to be zero. Minor of order 2 is obtained by taking any two rows and any two columns.

Minor of order ${{D}_{3}}=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{d}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{d}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{d}_{3}} \\ \end{matrix}\, \right|$. Minor of order 1 is every element of the matrix.

Rank of a matrix: The rank of a given matrix A is said to be $r$ if

(1) Every minor of A of order $r+1$ is zero.

(2) There is at least one minor of A of order $r$ which does not vanish. Here we can also say that the rank of a matrix A is said to be $r$, if (i) Every square submatrix of order $r+1$ is singular.

(ii) There is at least one square submatrix of order $r$ which is non-singular.

The rank $r$  of matrix A is written as $\rho (A)=r$.

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