RIGHTARROW \left[ {\BEGIN{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&{{b_2}}&{{c_2}}\\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right]\;\left[ {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right] = \;\left[ {\begin{array}{*{20}{c}} {{d_1}}\\ {{d_2}}\\ {{d_3}} \end{array}} \right]\)⇒ AX = B⇒ X = A - 1 B = \(\frac{{{\rm{adj\;}}\left( {\rm{A}} \right)}}{{\det {\rm{\;}}({\rm{A}})}}\;B\)⇒ If det (A) ≠ 0, system is consistent having unique solution.⇒ If det (A) = 0 and (adj A). B = 0, system is consistent, with infinitely many solutions.⇒ If det (A) = 0 and (adj A). B ≠ 0, system is inconsistent (no solution)If \(A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right]\) then determinant of A is given by:|A| = a11 × {(a22 × a33) - (a23 × a32)} - a12 × {(a21 × a33) - (a23 × a31)} + a13 × {(a21 × a32) - (a22 × a31)}Calculation:Given: The system of equations2x + y - 3z = 53x - 2y + 2z = 5 and5x - 3y - z = 16So, A = \(\left[ {\begin{array}{*{20}{c}} 2&1&{ - 3}\\ 3&{ - 2}&2\\ 5&{ - 3}&{ - 1} \end{array}} \right]\)det (A) = |A| = 2 × {( - 2 × - 1) - ( - 3 × 2)} - 1 × {(3 × - 1) - (2 × 5)} + ( - 3) × {(3 × - 3) - (5 × - 2)}⇒ |A| = 2 × (2 + 6) - 1 × ( - 3 - 10) - 3 × ( - 9 + 10)⇒ |A| = 16 + 13 - 3 = 26∴ |A| ≠ 0So, system is consistent having unique solution.