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Determinants

Short Answer Type

21. Find the value of the determinant

open vertical bar table row cell negative 1 end cell cell space space space space 6 end cell cell space space minus 2 end cell row 2 cell space space 1 end cell cell space space space space space 1 end cell row 4 cell space space 1 end cell cell space space minus 3 end cell end table close vertical bar

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22.

Find space the space value space of space the space determinant open vertical bar table row 5 cell space space space space 1 end cell cell space space space space space space 0 end cell row 2 cell space space space space 3 end cell cell space space minus 1 end cell row cell negative 3 end cell cell space space space space 2 end cell cell space space space space 0 end cell end table close vertical bar space

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23.

Find space the space value space of space the space determinant open vertical bar table row 212 cell space space space space 117 space space end cell cell space space 345 end cell row 19 cell space space 9 end cell cell space space 34 end cell row 7 cell space space 3 end cell cell space space 5 end cell end table close vertical bar

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24. Evaluate the following determinants :

open square brackets table row 12 cell space space 3 end cell cell space space 7 end cell row 13 cell space space 17 end cell cell space space 5 end cell row 15 cell space space 20 end cell cell space 12 end cell end table close square brackets squared

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25. Evaluate the following determinants :

open vertical bar table row straight a cell space straight h end cell cell space space straight g end cell row straight h cell space space straight b end cell cell space space space straight f end cell row straight g cell space space straight f end cell cell space space straight c end cell end table close vertical bar

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26. Prove that the determinant

open vertical bar table row straight x cell space space space space sin space straight theta end cell cell space space space space space cos space straight theta end cell row cell negative sin space straight theta end cell cell negative straight x end cell 1 row cell cos space straight theta end cell cell space space 1 end cell straight x end table close vertical bar

is independent of ө.

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Long Answer Type

27.

Evaluate space the space determinant increment space space equals space space open vertical bar table row 1 cell space space space space sin space straight theta end cell 1 row cell negative sin end cell cell space 1 end cell cell space space sin space straight theta end cell row cell negative 1 end cell cell negative sin end cell 1 end table close vertical bar space comma space Also comma space prove space that space 2 space less than space straight A space less than space 4.

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Short Answer Type

28.

space Evaluate increment space space equals space open vertical bar table row 0 cell sin space straight alpha end cell cell negative cos space straight alpha end cell row cell negative sin space straight alpha end cell 0 cell sin space straight beta end cell row cell cos space straight alpha end cell cell negative sin space straight beta end cell 0 end table close vertical bar

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29.

space Prove space that space open vertical bar table row cell straight b plus straight c end cell cell space space straight a end cell cell space space straight a end cell row straight b cell space space straight c plus straight a end cell cell space space straight b end cell row straight c cell space straight c end cell cell space space straight a plus straight b end cell end table close vertical bar space equals 4 abc

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30. $Prove space that open vertical bar table row straight a straight b straight c row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space equals straight a cubed plus straight b cubed plus straight c cubed space minus 3 abc$

Let space space space increment space space equals space open vertical bar table row straight a straight b straight c row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space space space space space space space space space space space space space space equals space straight a space space open vertical bar table row straight a straight b straight c row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space minus straight b space space open vertical bar table row straight a straight b straight c row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space plus straight c space space open vertical bar table row straight a straight b straight c row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space space space space space space space space space space space space space space equals space straight a space open square brackets table row cell left parenthesis straight b minus straight c right parenthesis end cell cell left parenthesis straight a plus straight b right parenthesis minus end cell cell left parenthesis straight c plus straight a right parenthesis end cell cell left parenthesis straight c minus straight a right parenthesis end cell end table close square brackets space minus straight b space open square brackets table row cell left parenthesis straight a minus straight b right parenthesis end cell cell left parenthesis straight a plus straight b right parenthesis minus end cell cell left parenthesis straight b plus straight c right parenthesis end cell cell left parenthesis straight c minus straight a right parenthesis end cell end table close square brackets space plus straight c space space open square brackets table row cell left parenthesis straight a minus straight b right parenthesis end cell cell left parenthesis straight c plus straight a right parenthesis minus end cell cell left parenthesis straight b minus straight c right parenthesis end cell cell left parenthesis straight b plus straight c right parenthesis end cell end table close square brackets space space space space space space space space space space space space space space equals space straight a space open square brackets straight a table row cell straight b plus straight b squared end cell cell negative ca minus bc end cell cell straight c squared plus ca minus ca plus end cell cell straight a squared end cell end table close square brackets space minus straight b space open square brackets straight a squared space minus straight b to the power of 2 space end exponent minus bc plus ab minus straight c squared plus ca close square brackets plus straight c open square brackets table row cell ac plus straight a squared end cell end table minus bc minus ab minus straight b squared space plus straight c squared close square brackets space space space space space space space space space space space space space space equals space straight a space open square brackets straight a squared table row cell plus straight b squared end cell cell negative straight c squared minus ab minus bc minus ca end cell plus cell straight a squared end cell end table close square brackets space minus straight b space open square brackets straight a squared space minus straight b to the power of 2 space end exponent minus straight c squared plus ab minus bc plus ca close square brackets plus straight c open square brackets table row cell straight a squared minus end cell end table straight b squared plus straight c squared minus ab minus bc plus ca close square brackets space space space space space space space space space space space space space space equals space space straight a cubed space plus ab squared space minus ac squared space plus straight a squared space straight b minus abc minus straight a squared space straight c minus straight a squared space straight b plus straight b cubed space plus straight b squared space straight c minus ab squared space plus bc squared space plus bc cubed space minus abc plus straight a squared space straight c space minus straight b squared straight c space plus straight c cubed space minus abc minus bc squared space plus straight c to the power of 2 space end exponent straight a space space space space space space space space space space space space space space equals space space straight a cubed space plus straight b cubed space plus straight c cubed space minus 3 abc

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