# JEE Main & Advanced Mathematics Determinants & Matrices Application of Determinants in Solving a System of Linear Equations

Application of Determinants in Solving a System of Linear Equations

Category : JEE Main & Advanced

(1) Solution of system of linear equations in three variables by Cramer's rule : The solution of the system of linear equations  ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z={{d}_{1}}$                       .....(i)

${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z={{d}_{2}}$                       .....(ii)

${{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z={{d}_{3}}$                       .....(iii)

Is given by $x=\frac{{{D}_{1}}}{D},\,\,\,\,\,\,y=\frac{{{D}_{2}}}{D}$ and $z=\frac{{{D}_{3}}}{D}$,

where, $D=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix}\, \right|\,,$        ${{D}_{1}}=\left| \,\begin{matrix} {{d}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{d}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{d}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix}\, \right|$

${{D}_{2}}=\left| \,\begin{matrix} {{a}_{1}} & {{d}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{d}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{d}_{3}} & {{c}_{3}} \\ \end{matrix}\, \right|\,,$ and ${{D}_{3}}=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{d}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{d}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{d}_{3}} \\ \end{matrix}\, \right|$

Provided that $D\ne 0$

(2) Conditions for consistency : For a system of 3 simultaneous linear equations in three unknown variable.

(i) If $D\ne 0$, then the given system of equations is consistent and has a unique solution given by $x=\frac{{{D}_{1}}}{D},\,\,\,y=\frac{{{D}_{2}}}{D}$ and $z=\frac{{{D}_{3}}}{D}$

(ii) If $D=0$ and ${{D}_{1}}={{D}_{2}}={{D}_{3}}=0$, then the given system of equations is consistent with infinitely many solutions.

(iii) If $D=0$ and at least one of the determinants ${{D}_{1}},\,\,{{D}_{2}},\,\,{{D}_{3}}$ is non-zero, then given of equations is inconsistent.

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