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Determinants

Question
CBSEENMA12034375

Without expanding, prove that

open vertical bar table row cell straight a plus straight l end cell straight m straight n row straight I cell straight a plus straight m end cell straight n row straight I straight m cell straight a plus straight n end cell end table close vertical bar space equals a squared space left parenthesis a plus l plus m plus n right parenthesis

Solution
Let space increment space equals space open vertical bar table row cell straight a plus straight l end cell straight m straight n row straight I cell straight a plus straight m end cell straight n row straight I straight m cell straight a plus straight n end cell end table close vertical bar space
space space space space space space space space space space space equals space space open vertical bar table row cell straight a plus straight l plus straight m plus straight n end cell straight m straight n row cell straight a plus straight l plus straight m plus straight n end cell cell space straight a plus straight m end cell straight n row cell straight a plus straight l plus straight m plus straight n end cell straight m cell space straight a plus straight n end cell end table close vertical bar space space comma space b y space C subscript 1 plus C subscript 2 space plus C subscript 3
space space space space space space space space space space space equals space left parenthesis straight a plus straight l plus straight m plus straight n right parenthesis space space open vertical bar table row 1 straight m cell space space space straight n end cell row 1 cell space space straight a plus straight m end cell cell space space straight n end cell row 1 straight m cell space straight a plus straight n end cell end table close vertical bar
space space space space space space space space space space space equals space space left parenthesis straight a plus straight l plus straight m plus straight n right parenthesis space space open vertical bar table row 1 cell space space space space straight m end cell cell space space space space space straight n end cell row 0 cell space space space space straight a end cell cell space space space space 0 end cell row 0 cell space space space 0 end cell cell space space space straight a end cell end table close vertical bar space comma space b y space straight R subscript 2 minus straight R subscript 1 comma space straight R subscript 3 minus straight R subscript 1
space space space space space space space space space space space equals space space left parenthesis straight a plus straight l plus straight m plus straight n right parenthesis space open square brackets table row cell left parenthesis 1 right parenthesis left parenthesis straight a right parenthesis left parenthesis straight a right parenthesis end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space open square brackets table row cell product space of space diagonal space elements end cell end table close square brackets
space space space space space space space space space space space equals space straight a squared space space space left parenthesis straight a plus straight l plus straight m plus straight n right parenthesis space

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