Question

The value of the determinant remains unchanged if its rows and columns are interchanged. $Verify space the space property space for space space increment space equals space open vertical bar table row 2 cell space space minus 3 end cell cell space space 5 end cell row 6 cell space space space 0 end cell cell space space 4 end cell row 1 cell space space space 5 end cell cell space 7 end cell end table close vertical bar$

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Solution

space increment space equals space open vertical bar table row 2 cell space space minus 3 end cell cell space space 5 end cell row 6 cell space space space 0 end cell cell space space 4 end cell row 1 cell space space space 5 end cell cell space 7 end cell end table close vertical bar Expanding space the space determinant space along space first space row comma space we space have increment space space equals 2 space open vertical bar table row 0 cell space space space space space space 4 end cell row 5 cell space space minus 7 end cell end table close vertical bar minus left parenthesis negative 3 right parenthesis space open vertical bar table row 6 cell space space space space 4 end cell row 1 cell space space space minus 7 end cell end table close vertical bar space plus 5 space open vertical bar table row 6 cell space space 0 end cell row 1 cell space space 5 end cell end table close vertical bar space space space space space equals space 2 left parenthesis 0 minus 20 right parenthesis plus 3 left parenthesis negative 42 minus 4 right parenthesis plus 5 left parenthesis 30 minus 0 right parenthesis equals negative 40 minus 138 plus 150 equals negative 28 By space interchanging space rows space and space columns comma space we space get increment subscript 1 space equals space space space space open vertical bar table row 2 cell space space minus 3 end cell cell space space 5 end cell row 6 cell space space space 0 end cell cell space space 4 end cell row 1 cell space space space 5 end cell cell space 7 end cell end table close vertical bar space equals 2 space space open vertical bar table row 0 cell space space space space space space 4 end cell row 5 cell space space minus 7 end cell end table close vertical bar minus left parenthesis negative 3 right parenthesis space open vertical bar table row 6 cell space space space space 4 end cell row 1 cell space space space minus 7 end cell end table close vertical bar space plus 5 space open vertical bar table row 6 cell space space 0 end cell row 1 cell space space 5 end cell end table close vertical bar space space space space space equals 2 space left parenthesis 0 minus 20 right parenthesis plus 3 left parenthesis negative 42 minus 4 right parenthesis plus 5 left parenthesis 30 minus 0 right parenthesis space space space space space equals negative 40 minus 138 plus 150 equals negative 28 therefore space space space increment space equals space increment subscript 1 space Hence space property space is space verified.

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