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Write the minors and cofactors of the elements of following determinants:
$open vertical bar table row 1 cell space space space 0 end cell 4 row 3 cell space space 5 end cell cell negative 1 end cell row 0 cell space 1 end cell cell space 2 end cell end table close vertical bar$

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Long Answer Type

152. Find the minors and cofactors of the elements of the determinant:

open vertical bar table row 2 cell space space space minus 3 end cell cell space space space 5 end cell row 6 cell space space 0 end cell cell space space space 4 end cell row 1 cell space space 5 end cell cell negative 7 end cell end table close vertical bar

and verify that a₁₁A₃₁ + a₁₃A₃₃ = 0.

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Short Answer Type

153. Write the minors and cofactors of the elements of following determinants:

open vertical bar table row 1 cell space space 0 end cell cell space space 0 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 0 cell space 0 end cell cell space space 1 end cell end table close vertical bar

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154.

Using cofactors of elements of second row, evaluate:
$increment space equals space open vertical bar table row 5 cell space space 3 end cell cell space space 8 end cell row 2 cell space space 0 end cell cell space space 1 end cell row 1 cell space 2 end cell cell space space 3 end cell end table close vertical bar$

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155.
Using cofactors of elements of third column, evaluate:
$increment space equals space open vertical bar table row 1 cell space space straight x end cell cell space space yz end cell row 1 cell space space straight y end cell cell space space zx end cell row 1 cell space space straight z end cell cell space space xy end cell end table close vertical bar.$

increment space equals space open vertical bar table row 1 cell space space straight x end cell cell space space yz end cell row 1 cell space space straight y end cell cell space space zx end cell row 1 cell space space straight z end cell cell space space xy end cell end table close vertical bar

straight A subscript 13 space equals space left parenthesis negative 1 right parenthesis to the power of 1 plus 3 end exponent space open vertical bar table row 1 cell space space space space straight y end cell row 1 cell space space space space straight z end cell end table close vertical bar space equals space 1 left parenthesis straight z minus straight y right parenthesis space equals space straight z minus straight y straight A subscript 23 space equals space left parenthesis negative 1 right parenthesis to the power of 2 plus 3 end exponent open vertical bar table row 1 cell space space space straight x end cell row 1 cell space space space straight z end cell end table close vertical bar space equals space minus left parenthesis straight z minus straight x right parenthesis space equals space straight x minus straight z straight A subscript 33 space equals space left parenthesis negative 1 right parenthesis to the power of 3 plus 3 end exponent space open vertical bar table row 1 cell space space space straight x end cell row 1 cell space space straight y end cell end table close vertical bar space equals space 1 left parenthesis straight y minus straight x right parenthesis space equals space straight y minus straight x

Now comma space increment space equals space straight a subscript 13 straight A subscript 13 plus straight a subscript 23 straight A subscript 23 plus straight a subscript 33 straight A subscript 33 space space space space space equals space yz left parenthesis straight z minus straight y right parenthesis plus zx left parenthesis straight x minus straight z right parenthesis plus xy left parenthesis straight y minus straight x right parenthesis space space space space space equals space yz squared minus straight y squared straight z plus zx squared minus straight z squared straight x plus xy squared minus straight x squared straight y space space space space space equals space left parenthesis xy squared minus xz squared right parenthesis plus left parenthesis yz squared minus yx squared right parenthesis plus left parenthesis zx squared minus zy squared right parenthesis space space space space space space equals straight x left parenthesis straight y squared minus straight z squared right parenthesis space plus straight y left parenthesis straight z squared minus straight x squared right parenthesis space plus space straight z left parenthesis straight x squared minus straight y squared right parenthesis therefore space space space space space increment space equals space left parenthesis straight x minus straight y right parenthesis thin space left parenthesis straight y minus straight z right parenthesis thin space left parenthesis straight z minus straight x right parenthesis thin space space space space open square brackets because space sum from blank to blank of straight a space left parenthesis straight b squared minus straight c squared right parenthesis space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space close square brackets

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Multiple Choice Questions

156. If

increment space equals open vertical bar table row cell straight a subscript 11 end cell cell space space straight a subscript 12 end cell cell space straight a subscript 13 end cell row cell straight a subscript 21 end cell cell straight a subscript 22 end cell cell space straight a subscript 23 end cell row cell straight a subscript 31 end cell cell space straight a subscript 32 end cell cell space straight a subscript 33 end cell end table close vertical bar

and A_ij is Cofactors of a_ij, then value of ∆ is given by

a₁₁A₃₁ + a₁₂A₃₂ + a₁₃A₃₃
a₁₁A₁₁ + a₁₂A₂₁ + a₁₃A₃₁
a₂₁A₁₁ + a₂₂A₁₂ + a₂₃A₁₃
a₂₁A₁₁ + a₂₂A₁₂ + a₂₃A₁₃

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Short Answer Type

157.

If A, B, C are square matrices of the same order such that AB = BA = I and AC = CA = I then B = C.

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158.

Find adj A for $straight A space equals space open square brackets table row 2 cell space space space 3 end cell row 1 cell space space space 4 end cell end table close square brackets.$

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159.

Find the adjoint of the matrix $open square brackets table row 1 cell space space space space space 2 end cell row 3 cell space space space space 4 end cell end table close square brackets.$