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Determinants

Question
CBSEENMA12034395
Wired Faculty App

show space that open vertical bar table row cell straight a plus straight b plus nc end cell cell left parenthesis straight n minus 1 right parenthesis straight a end cell cell left parenthesis straight n minus 1 right parenthesis end cell row cell left parenthesis straight n minus 1 right parenthesis space straight c end cell cell straight b plus straight c plus na end cell cell left parenthesis straight n minus 1 right parenthesis end cell row cell left parenthesis straight n minus 1 right parenthesis straight c end cell cell left parenthesis straight n minus 1 right parenthesis straight a end cell cell straight c plus straight a plus nb end cell end table close vertical bar space equals straight n left parenthesis straight a plus straight b plus straight c right parenthesis cubed

Solution
Let space space space space increment space equals open vertical bar table row cell straight a plus straight b plus nc end cell cell left parenthesis straight n minus 1 right parenthesis straight a end cell cell left parenthesis straight n minus 1 right parenthesis end cell row cell left parenthesis straight n minus 1 right parenthesis space straight c end cell cell straight b plus straight c plus na end cell cell left parenthesis straight n minus 1 right parenthesis end cell row cell left parenthesis straight n minus 1 right parenthesis straight c end cell cell left parenthesis straight n minus 1 right parenthesis straight a end cell cell straight c plus straight a plus nb end cell end table close vertical bar space space space space space space space space space space space space space space space space equals space open vertical bar table row cell straight n left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell na plus straight a end cell cell nb minus straight b end cell row cell straight n left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell space space straight b plus straight c plus nb end cell cell nb minus straight b end cell row cell straight n left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell na minus straight a end cell cell space space straight c plus straight a plus nb end cell end table close vertical bar space space by space straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3 space space space space space space space space space space space space space space equals space straight n left parenthesis straight a plus straight b plus straight c right parenthesis space space open vertical bar table row 1 cell straight n space straight a minus straight a end cell cell nb minus straight b end cell row 1 cell space space space space straight b plus straight c plus nb end cell cell nb minus straight b end cell row 1 cell na minus straight a end cell cell space space straight c plus straight a plus nb end cell end table close vertical bar space space comma space by space taking space space common space form space straight C subscript 1 space space space space space space space space space space space space space equals space straight n left parenthesis straight a plus straight b plus straight c right parenthesis space space open vertical bar table row 1 cell space space na minus straight a end cell cell nb minus straight b end cell row 0 cell space space space space straight a plus straight b plus straight c end cell 0 row 0 0 cell space straight a plus straight b plus straight c end cell end table close vertical bar space space space space space space comma space by space straight R subscript 2 minus straight R subscript 1 comma straight R subscript 3 minus straight R subscript 1 space space space space space space space space space space space space space equals space straight n left parenthesis straight a plus straight b plus straight c right parenthesis space space open square brackets table row cell 1. left parenthesis straight a plus straight b plus straight c right parenthesis space left parenthesis straight a plus straight b plus straight c right parenthesis end cell end table close square brackets space space open square brackets table row cell product space of space diagonal space element space straight s end cell end table close square brackets space space space space space space space space space space space space space equals space straight n left parenthesis straight a plus straight b plus straight c right parenthesis cubed space space

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