Without expansion, show that

$open vertical bar table row 0 cell space space space straight a end cell cell space space minus straight b end cell row cell negative straight a end cell cell space space 0 end cell cell space space straight c end cell row straight b cell space space straight c end cell cell space space 0 end cell end table close vertical bar space equals space 0$

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Solution

Let space space increment equals space open vertical bar table row 0 cell space space space straight a end cell cell space space minus straight b end cell row cell negative straight a end cell cell space space 0 end cell cell space space straight c end cell row straight b cell space space straight c end cell cell space space 0 end cell end table close vertical bar space space space space space space space space space space space space equals space left parenthesis negative 1 space right parenthesis cubed space space space open vertical bar table row 0 cell space space space straight a end cell cell space space minus straight b end cell row cell negative straight a end cell cell space space 0 end cell cell space space straight c end cell row straight b cell space space straight c end cell cell space space 0 end cell end table close vertical bar space comma space by space taking minus 1 space common space from space each space of space three space rows space space space space space space space space space space space space equals space space minus space space space space open vertical bar table row 0 cell space space space straight a end cell cell space space minus straight b end cell row cell negative straight a end cell cell space space 0 end cell cell space space straight c end cell row straight b cell space space straight c end cell cell space space 0 end cell end table close vertical bar space space comma space by space interchanging space rows space and space columns. space space space space space space space space space space space space space equals space minus space increment therefore space space space space increment space equals space minus space increment space space space space rightwards double arrow 2 increment space equals 0 space rightwards double arrow space space space increment space space space space equals 0 rightwards double arrow space space space space space open vertical bar table row 0 cell space space space straight a end cell cell space space minus straight b end cell row cell negative straight a end cell cell space space 0 end cell cell space space straight c end cell row straight b cell space space straight c end cell cell space space 0 end cell end table close vertical bar space equals 0

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