8.2 Rank of a Matrix

  Matrices and Determinants

• It has atleast minors of order $r$ is different from zero
• All minors of $A$ of order higher than $r$ are zero
• The rank of $A$ is denoted by $\rho (A)$

Note:-

• The rank of a zero matrix is 0 and the rank of an identity matrix of order $n$ is $n$.
• The rank of a matrix in echelon form is equal to the number of non zero rows of the matrix .
• The rank of a non -singular matrix of order $n$ is $n$.

Solution of Simultaneous Linear Equations by Rank Method

Let us consider a system of n linear equations in n unknowns say ${x_1},{x_2},............,{x_n}$ given as below

$\begin{equation} \begin{aligned} {a_{11}}{x_1} + {a_{12}}{x_2} + ............. + {a_{1n}}{x_n} = {b_1} \\ {a_{21}}{x_1} + {a_{22}}{x_2} + ............. + {a_{2n}}{x_n} = {b_2} \\ ........................................... \\ ......................................... \\ {a_{n1}}{x_1} + {a_{n2}}{x_2} + ............. + {a_{nn}}{x_n} = {b_n} \\\end{aligned} \end{equation} $

We write above system of equations in the matrix form

$\left[ {\begin{array}{c} {{a_{11}}{x_1} + {a_{12}}{x_2} + .... + {a_{1n}}{x_n}} \\ {{a_{21}}{x_1} + {a_{22}}{x_2} + .... + {a_{2n}}{x_n}} \\ {.....................................} \\ {{a_{n1}}{x_1} + {a_{n2}}{x_2} + .... + {a_{nn}}{x_n}} \end{array}} \right] = $$\left[ {\begin{array}{c} {{b_1}} \\ {{b_2}} \\{....} \\ {{b_n}} \end{array}} \right]$

$ \Rightarrow $ $\left[ {\begin{array}{c} {{a_{11}}}&{{a_{12}}}&{....}&{{a_{1n}}} \\ {{a_{21}}}&{{a_{22}}}&{....}&{{a_{2n}}} \\ {....}&{....}&{....}&{....} \\ {{a_{n1}}}&{{a_{n2}}}&{....}&{{a_{nn}}} \end{array}} \right]\left[ {\begin{array}{c} {{x_1}} \\ {{x_2}} \\ {....} \\ {{x_n}} \end{array}} \right] = $$\left[ {\begin{array}{c} {{b_1}} \\ {{b_2}} \\ {....} \\ {{b_n}} \end{array}} \right]$

$ \Rightarrow $ $AX = B$

and $C = \left[ {A:B} \right]$$ = \left[ {\begin{array}{c} {{a_{11}}}&{{a_{12}}}&{....}&{{a_{1n}}} \\ {{a_{21}}}&{{a_{22}}}&{....}&{{a_{2n}}} \\ {....}&{....}&{....}&{....} \\ {{a_{n1}}}&{{a_{n2}}}&{....}&{{a_{nn}}} \end{array}} \right.\left. {\begin{array}{c} {{b_1}} \\ {{b_2}} \\ {....} \\ {{b_n}}\end{array}} \right]$

Where $C$ is called the augmented matrix.