JEE Main & Advanced Mathematics Determinants 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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