Expand the following determinant:
   $| {\begin{array}{*{20}{c}}
  1&{ - 3}&4
  3&5&{ - 3}
  2&{ - 5}&0
\end{array}} \right| $

Question

Expand the following determinant:   $| {\begin{array}{*{20}{c}}  1&{ - 3}&4    3&5&{ - 3}    2&{ - 5}&0 \end{array}} \right| $

Vedantu Content Team · Accepted Answer

\[  {\text{Let }}\left| {\begin{array}{*{20}{c}}  a&b&c \\   d&e&f \\   g&h&i \end{array}} \right|{\text{ be a general determinant                                                                }}   \\  {\text{As we know that }}\left| {\begin{array}{*{20}{c}}  a&b&c \\   d&e&f \\   g&h&i \end{array}} \right|{\text{ is expanded as,}}   \\   \Rightarrow \left| {\begin{array}{*{20}{c}}  a&b&c \\   d&e&f \\   g&h&i \end{array}} \right|{\text{ }} = a\left| {\begin{array}{*{20}{c}}  e&f \\   h&i \end{array}} \right| - b\left| {\begin{array}{*{20}{c}}  d&f \\   g&i \end{array}} \right| + c\left| {\begin{array}{*{20}{c}}  d&e \\   g&h \end{array}} \right|{\text{                                                                        }}   \\  {\text{This can be reduced as,}}   \\   \Rightarrow a\left| {\begin{array}{*{20}{c}}  e&f \\   h&i \end{array}} \right| - b\left| {\begin{array}{*{20}{c}}  d&f \\   g&i \end{array}} \right| + c\left| {\begin{array}{*{20}{c}}  d&e \\   g&h \end{array}} \right|{\text{   =    }}a\left( {ei - hf} \right) - b\left( {di - gf} \right) + c\left( {dh - ge} \right){\text{       }}   \\   \Rightarrow {\text{So,    }}\left| {\begin{array}{*{20}{c}}  a&b&c \\   d&e&f \\   g&h&i \end{array}} \right|{\text{ }} = {\text{ }}a\left( {ei - hf} \right) - b\left( {di - gf} \right) + c\left( {dh - ge} \right){\text{                                                      (1)}}   \\  {\text{Now we know we have to expand  }}\left| {\begin{array}{*{20}{c}}  1&{ - 3}&4 \\   3&5&{ - 3} \\   2&{ - 5}&0 \end{array}} \right|{\text{                                                   }}   \\  {\text{So, to expand }}\left| {\begin{array}{*{20}{c}}  1&{ - 3}&4 \\   3&5&{ - 3} \\   2&{ - 5}&0 \end{array}} \right|{\text{ first we have to compare its elements with the elements of  }}\left| {\begin{array}{*{20}{c}}  a&b&c \\   d&e&f \\   g&h&i \end{array}} \right|{\text{ }}.   \\  {\text{So on comparing we get }}a = 1,b =  - 3,c = 4,d = 3,e = 5,f =  - 3,g = 2,h =  - 5{\text{ and }}i = 0   \\  {\text{Now putting values of }}a,b,c,d,e,f,g,h{\text{ and }}i{\text{ in equation 3 we get,}}   \\   \Rightarrow \left| {\begin{array}{*{20}{c}}  1&{ - 3}&4 \\   3&5&{ - 3} \\   2&{ - 5}&0 \end{array}} \right|{\text{  }} = {\text{ }}1\left( {5*0 - ( - 5)*( - 3)} \right) + 3\left( {3*0 - 2*( - 3)} \right) + 4\left( {3*( - 5) - 2*5} \right)   \\   \Rightarrow {\text{So, }}\left| {\begin{array}{*{20}{c}}  1&{ - 3}&4 \\   3&5&{ - 3} \\   2&{ - 5}&0 \end{array}} \right|{\text{  =  }} - 15 + 18 - 100 =  - 97   \\  {\text{Hence  }}\left| {\begin{array}{*{20}{c}}  1&{ - 3}&4 \\   3&5&{ - 3} \\   2&{ - 5}&0 \end{array}} \right|{\text{  }} =  - 97{\text{ }}   \\  {\text{NOTE: -  Whenever you came up to expand a determinant then better way is to expand using cofactors}}{\text{.}}   \\  {\text{While expanding calculations should be carefully done}}{\text{.}}   \\ \]