Evaluate from Mathematics Determinants Class 12 Jammu and Kashm

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Determinants

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Mathematics Part I

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Mathematics

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Determinants

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If A= $open square brackets table row 1 cell space space 2 end cell row 4 cell space space 2 end cell end table close square brackets comma space then space show space that space left enclose 2 space straight A space open vertical bar space equals 4 space close vertical bar space right enclose straight A space end enclose end enclose$

straight A space equals space open square brackets table row 1 cell space space 2 end cell row 4 cell space space 2 end cell end table close square brackets therefore space space open vertical bar space straight A space close vertical bar space equals space space space open square brackets table row 1 cell space space 2 end cell row 4 cell space space 2 end cell end table close square brackets space equals space left parenthesis 1 right parenthesis left parenthesis 2 right parenthesis minus left parenthesis 2 right parenthesis left parenthesis 4 right parenthesis equals 2 minus 8 equals negative 6 space space space space space space 2 space straight A space space equals space space open square brackets table row 2 cell space space 4 end cell row 8 cell space space 4 end cell end table close square brackets therefore space space open vertical bar space 2 space straight A space close vertical bar space equals space space open square brackets table row 2 cell space space 4 end cell row 8 cell space space 4 end cell end table close square brackets space equals 8 minus 32 equals space minus 24 equals 4 left parenthesis negative 6 right parenthesis equals space 4 space open vertical bar space straight A space close vertical bar therefore space space open vertical bar space 2 space straight A space close vertical bar space equals space 4 space open vertical bar space straight A space close vertical bar

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left parenthesis straight i right parenthesis open square brackets table row cell cos space straight theta end cell cell negative sin space straight theta space end cell row cell sin space straight theta end cell cell cos space straight theta end cell end table close square brackets space left parenthesis ii right parenthesis space open square brackets table row cell straight x squared minus straight x plus 1 end cell cell straight x minus 1 end cell row cell straight x plus 1 end cell cell straight x plus 1 end cell end table close square brackets

left parenthesis straight i right parenthesis space Let space space increment space equals space open square brackets table row cell cos space straight theta end cell cell negative sin space straight theta space end cell row cell sin space straight theta end cell cell cos space straight theta end cell end table close square brackets space space equals space cos squared space straight theta space plus sin space squared space straight theta space equals 1 left parenthesis i i right parenthesis space Let space increment equals space space open square brackets table row cell straight x squared minus straight x plus 1 end cell cell straight x minus 1 end cell row cell straight x plus 1 end cell cell straight x plus 1 end cell end table close square brackets space equals space left parenthesis straight x squared space minus straight x plus 1 right parenthesis space left parenthesis straight x plus 1 right parenthesis minus left parenthesis straight x plus 1 right parenthesis space space space space space space space space space space space space space space space space space equals space space straight x to the power of 3 space end exponent minus straight x squared space plus straight x plus straight x squared space minus straight x plus 1 equals straight x to the power of 3 space space end exponent minus straight x squared space plus 2

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Evaluate the determinants

open vertical bar table row 3 cell space minus 4 end cell cell space space 5 end cell row 1 cell space space space 1 end cell cell space minus 2 end cell row 2 cell space space space 3 end cell cell space 1 end cell end table close vertical bar

Let space space space increment space equals space open square brackets table row 3 cell space space minus 4 end cell cell space space space space 5 end cell row 1 cell space space space space 1 end cell cell space minus 2 end cell row 2 cell space space space space 3 end cell cell space space space 1 end cell end table close square brackets space space equals 3 space open vertical bar table row 1 cell space space minus 2 end cell row 3 cell space space space space 1 end cell end table close vertical bar space minus left parenthesis negative 4 right parenthesis space space open vertical bar table row 1 cell space space minus 2 end cell row 3 cell space space space space 1 end cell end table close vertical bar space plus 5 space open vertical bar table row 1 cell space space space space space 0 end cell row 2 cell space space space space space 3 end cell end table close vertical bar space space space space space space space space space space space space space equals space 3 left parenthesis 1 plus 6 right parenthesis plus 4 left parenthesis 1 plus 45 right parenthesis plus 5 left parenthesis 3 minus 2 right parenthesis space space space space space space space space space space space space space equals space 3 space left parenthesis 7 right parenthesis plus 4 left parenthesis 5 right parenthesis plus 5 left parenthesis 1 right parenthesis equals 21 plus 20 plus 5 equals 46 space space space space space space space space

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Evaluate the determinants

open vertical bar table row 3 cell negative 1 end cell 2 row 0 cell space space 0 end cell cell negative 1 end cell row 3 cell space 5 end cell 0 end table close vertical bar

Let space space increment space equals space space open vertical bar table row 3 cell negative 1 end cell 2 row 0 cell space space 0 end cell cell negative 1 end cell row 3 cell space 5 end cell 0 end table close vertical bar space space space space space space space space space space space space equals space space minus 0 space open vertical bar table row cell negative 1 end cell cell space space minus 2 end cell row cell negative 5 end cell cell space space space 0 end cell end table close vertical bar plus 0 space open vertical bar table row 3 cell space space minus 2 end cell row 3 cell space space space space space space 0 end cell end table close vertical bar space minus left parenthesis negative 1 right parenthesis space open vertical bar table row 3 cell space space minus 2 end cell row 3 cell space space space 0 end cell end table close vertical bar space minus left parenthesis negative 1 right parenthesis space open vertical bar table row 3 cell space space minus 1 end cell row 3 cell space space minus 5 end cell end table close vertical bar space space space space space space space space space space space space equals space 0 plus 0 plus left parenthesis negative 15 plus 3 right parenthesis space equals negative 12 space space space

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$open vertical bar table row 2 cell space space space space space space 4 end cell row cell negative 5 end cell cell space space space minus 1 end cell end table close vertical bar$

Let space space space increment space equals space open vertical bar table row 2 cell space space space space space space 4 end cell row cell negative 5 end cell cell space space space minus 1 end cell end table close vertical bar space space space space space space space space space space space space space space equals space left parenthesis 2 right parenthesis space left parenthesis negative 1 right parenthesis minus left parenthesis negative 4 right parenthesis left parenthesis negative 5 right parenthesis space space space space space space space space space space space space space space equals space minus 2 plus 20 equals 18 space space space space space space

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