Prove that the determinant is independent of ө. from M

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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.

Let space space space space increment space space equals space 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 space space space space space space space space space space space space space space space equals straight x space space open vertical bar table row cell negative straight x end cell cell space space space 1 end cell row 1 cell space space space straight x end cell end table close vertical bar space minus sin space straight theta space open vertical bar table row cell negative sin space straight theta end cell cell space space 1 end cell row cell cos space straight theta end cell cell space space straight x end cell end table close vertical bar plus cos space straight theta space open vertical bar table row cell negative sin space straight theta end cell cell space minus straight x end cell row cell cos space straight theta end cell cell space space space space 1 end cell end table close vertical bar space space space space space space space space space space space space space space space equals straight x left parenthesis negative straight x squared minus 1 right parenthesis space minus sin space left parenthesis negative straight x space sin space straight theta space minus cos space straight theta right parenthesis minus cos space straight theta left parenthesis negative sin space straight theta plus cos space straight theta right parenthesis space space space space space space space space space space space space space space space equals negative straight x cubed minus straight x plus straight x space sin squared space straight theta space plus sin space straight theta space cos space straight theta space minus sin space straight theta space cos space straight theta space plus straight x space cos squared space straight theta space space space space space space space space space space space space space space space equals space minus straight x cubed minus straight x plus straight x left parenthesis sin squared space straight theta space plus cos squared space straight theta right parenthesis space equals space straight x cubed minus straight x space space space space space space space space space space space space space space space equals negative straight x cubed space comma space which space is space independent space of space space straight theta

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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

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