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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