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 \[\rho (A)=r\].

*play_arrow*Definition*play_arrow*Order of a Matrix*play_arrow*Equality of Matrices*play_arrow*Types of Matrices*play_arrow*Trace of a Matrix*play_arrow*Addition and Subtraction of Matrices*play_arrow*Scalar Multiplication of Matrices*play_arrow*Multiplication of Matrices*play_arrow*Positive Integral Powers of a Matrix*play_arrow*Transpose of a Matrix*play_arrow*Special Types of Matrices*play_arrow*Adjoint of a Square Matrix*play_arrow*Inverse of a Matrix*play_arrow*Rank of Matrix*play_arrow*Echelon Form of a Matrix*play_arrow*Homogeneous and Non-homogeneous Systems of Linear Equations*play_arrow*Consistency of a System of Linear Equation*play_arrow*Cayley-Hamilton Theorem*play_arrow*Geometrical Transformations*play_arrow*Matrices of Rotation of Axes

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