Concept:
Determinant:
To every square matrix A of order n, we can associate a number (real or complex) called determinant of the square matrix A and it is denoted by |A|
The determinant of a square matrix of order two:
If \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {{{\rm{a}}_{11}}}&{{{\rm{a}}_{12}}{\rm{\;}}}\\ {{{\rm{a}}_{21}}}&{{{\rm{a}}_{22}}} \end{array}} \right]\) is a square matrix of order two, then its determinant is given by,
\({\rm{|A|}} = \left| {\begin{array}{*{20}{c}} {{{\rm{a}}_{11}}}&{{{\rm{a}}_{12}}{\rm{\;}}}\\ {{{\rm{a}}_{21}}}&{{{\rm{a}}_{22}}} \end{array}} \right|\)
|A| = a11 a22 – a12 a21
The determinant of a square matrix of order four
If \({\rm{A = }}\left[ {\begin{array}{*{20}{c}} {{{\rm{a}}_{{\rm{11}}}}}&{{{\rm{a}}_{{\rm{12}}}}}&{{{\rm{a}}_{{\rm{13}}}}}&{{{\rm{a}}_{{\rm{14}}}}}\\ {{{\rm{a}}_{{\rm{21}}}}}&{{{\rm{a}}_{{\rm{22}}}}}&{{{\rm{a}}_{{\rm{23}}}}}&{{{\rm{a}}_{{\rm{24}}}}}\\ {{{\rm{a}}_{{\rm{31}}}}}&{{{\rm{a}}_{{\rm{32}}}}}&{{{\rm{a}}_{{\rm{33}}}}}&{{{\rm{a}}_{{\rm{34}}}}}\\ {{{\rm{a}}_{{\rm{41}}}}}&{{{\rm{a}}_{{\rm{42}}}}}&{{{\rm{a}}_{{\rm{43}}}}}&{{{\rm{a}}_{{\rm{44}}}}} \end{array}} \right]\) is a square matrix of order four, then its determinant is given by,
\({\rm{|A| = }}\left| {\begin{array}{*{20}{c}} {{{\rm{a}}_{{\rm{11}}}}}&{{{\rm{a}}_{{\rm{12}}}}}&{{{\rm{a}}_{{\rm{13}}}}}&{{{\rm{a}}_{{\rm{14}}}}}\\ {{{\rm{a}}_{{\rm{21}}}}}&{{{\rm{a}}_{{\rm{22}}}}}&{{{\rm{a}}_{{\rm{23}}}}}&{{{\rm{a}}_{{\rm{24}}}}}\\ {{{\rm{a}}_{{\rm{31}}}}}&{{{\rm{a}}_{{\rm{32}}}}}&{{{\rm{a}}_{{\rm{33}}}}}&{{{\rm{a}}_{{\rm{34}}}}}\\ {{{\rm{a}}_{{\rm{41}}}}}&{{{\rm{a}}_{{\rm{42}}}}}&{{{\rm{a}}_{{\rm{43}}}}}&{{{\rm{a}}_{{\rm{44}}}}} \end{array}} \right|\)
\(\left| {\rm{A}} \right|{\rm{ = }}{{\rm{a}}_{{\rm{11}}}}{\rm{\;}}{\left( {{\rm{ - 1}}} \right)^{{\rm{1 + 1}}}}\left| {\begin{array}{*{20}{c}} {{{\rm{a}}_{{\rm{22}}}}}&{{{\rm{a}}_{{\rm{23}}}}}&{{{\rm{a}}_{{\rm{24}}}}}\\ {{{\rm{a}}_{{\rm{32}}}}}&{{{\rm{a}}_{{\rm{33}}}}}&{{{\rm{a}}_{{\rm{34}}}}}\\ {{{\rm{a}}_{{\rm{42}}}}}&{{{\rm{a}}_{{\rm{43}}}}}&{{{\rm{a}}_{{\rm{44}}}}} \end{array}} \right|{\rm{ + }}{{\rm{a}}_{{\rm{12}}}}{\rm{\;}}{\left( {{\rm{ - 1}}} \right)^{{\rm{1 + 2}}}}{\rm{\;}}\left| {\begin{array}{*{20}{c}} {{{\rm{a}}_{{\rm{21}}}}}&{{{\rm{a}}_{{\rm{23}}}}}&{{{\rm{a}}_{{\rm{24}}}}}\\ {{{\rm{a}}_{{\rm{31}}}}}&{{{\rm{a}}_{{\rm{33}}}}}&{{{\rm{a}}_{{\rm{34}}}}}\\ {{{\rm{a}}_{{\rm{41}}}}}&{{{\rm{a}}_{{\rm{43}}}}}&{{{\rm{a}}_{{\rm{44}}}}} \end{array}} \right|\)\( + {a_{13}}\;{\left( { - 1} \right)^{1 + 3}}\;\left| {\begin{array}{*{20}{c}} {{a_{21}}}&{{a_{22}}}&{{a_{24}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{34}}}\\ {{a_{41}}}&{{a_{42}}}&{{a_{44}}} \end{array}} \right| + {a_{14}}\;{\left( { - 1} \right)^{1 + 4}}\;\left| {\begin{array}{*{20}{c}} {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}}\\ {{a_{41}}}&{{a_{42}}}&{{a_{43}}} \end{array}} \right|\)
Calculation:
Given matrix
\({\rm{A = }}{\rm{}}\left| {\begin{array}{*{20}{c}} {\rm{0}}&{\rm{1}}&{\rm{2}}&{\rm{3}}\\ {\rm{1}}&{\rm{0}}&{\rm{3}}&{\rm{0}}\\ {\rm{2}}&{\rm{3}}&{\rm{0}}&{\rm{1}}\\ {\rm{3}}&{\rm{0}}&{\rm{1}}&{\rm{2}} \end{array}} \right|\)
\(\left| {\rm{A}} \right|{\rm{ = 0\;}}{\left( {{\rm{ - 1}}} \right)^{{\rm{1 + 1}}}}\left| {\begin{array}{*{20}{c}} {\rm{0}}&{\rm{3}}&{\rm{0}}\\ {\rm{3}}&{\rm{0}}&{\rm{1}}\\ {\rm{0}}&{\rm{1}}&{\rm{2}} \end{array}} \right|{\rm{ + 1\;}}{\left( {{\rm{ - 1}}} \right)^{{\rm{1 + 2}}}}{\rm{\;}}\left| {\begin{array}{*{20}{c}} {\rm{1}}&{\rm{3}}&{\rm{0}}\\ {\rm{2}}&{\rm{0}}&{\rm{1}}\\ {\rm{3}}&{\rm{1}}&{\rm{2}} \end{array}} \right|{\rm{ + 2\;}}{\left( {{\rm{ - 1}}} \right)^{{\rm{1 + 3}}}}{\rm{\;}}\left| {\begin{array}{*{20}{c}} {\rm{1}}&{\rm{0}}&{\rm{0}}\\ {\rm{2}}&{\rm{3}}&{\rm{1}}\\ {\rm{3}}&{\rm{0}}&{\rm{2}} \end{array}} \right|{\rm{ + 3\;}}{\left( {{\rm{ - 1}}} \right)^{{\rm{1 + 4}}}}{\rm{\;}}\left| {\begin{array}{*{20}{c}} {\rm{1}}&{\rm{0}}&{\rm{3}}\\ {\rm{2}}&{\rm{3}}&{\rm{0}}\\ {\rm{3}}&{\rm{0}}&{\rm{1}} \end{array}} \right|\)
|A| = 0 + (-1) × [1 (0 - 1) - 3 (4 - 3) + 0] + 2 [1 (6 - 0) - 0 + 0] + (-3) [1 (3 - 0) - 0 + 3 (0 - 9)]
|A| = 0 - (-1 - 3) + 2 (6) – 3 (3 - 27)
|A| = 0 + 4 + 12 +72
|A| = 88
