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

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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 $open square brackets table row 1 cell space space space space space 2 end cell row 3 cell space space space space 4 end cell end table close square brackets.$

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Long Answer Type

160. Find the adjoint of the matrix
$open square brackets table row 1 cell space space space minus 1 end cell cell space space space space space space 2 end cell row 2 cell space space space space space 3 end cell cell space space space space space 5 end cell row cell negative 2 end cell cell space space space space space 0 end cell cell space space space space 1 end cell end table close square brackets.$

Let space straight A space equals space open square brackets table row 1 cell negative 1 end cell cell space space 2 end cell row 2 cell space space 3 end cell cell space space 5 end cell row cell negative 2 end cell cell space space 0 end cell cell space 1 end cell end table close square brackets therefore space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space space minus 1 end cell cell space space space space 2 space end cell row 2 cell space space space space space 3 end cell cell space space 5 end cell row cell negative 2 end cell cell space space space space space 0 end cell cell space space 1 end cell end table close vertical bar

Co-factors of the elements of first row of | A | are

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

i.e. 3 – 0, – (2 + 10), 0 + 6 i.e. 3, – 12, 6 respectively.
Co-factors of the elements of second row of | A | are

negative open vertical bar table row cell negative 1 end cell cell space space 2 end cell row 0 cell space space space 1 space space end cell end table close vertical bar comma space space open vertical bar table row 1 cell space space space space 2 end cell row cell negative 2 end cell cell space space space 1 end cell end table close vertical bar comma space space minus open vertical bar table row 1 cell space space space space minus 1 end cell row cell negative 2 end cell cell space space space space space space space 0 end cell end table close vertical bar

i.e. – (– 1 – 0), 1 + 4, – (0 – 2) i.e. 1, 5, 2 respectively.
Cofactors of the elements of third row of | A | are
$open vertical bar table row cell negative 1 end cell cell space space space space 2 end cell row 3 cell space space space 5 end cell end table close vertical bar comma space space space space space minus open vertical bar table row 1 cell space space space space 2 end cell row 2 cell space space space space 5 end cell end table close vertical bar comma space space space open vertical bar table row 1 cell space space space space minus 1 end cell row 2 cell space space space space space space space 3 end cell end table close vertical bar$
i.e. – 5 – 6, – (5 – 4), 3 + 2
∴ –11,–1, 5 respectively.
$therefore space space space adj. space straight A space equals space open square brackets table row 3 cell space space minus 12 end cell cell space space space 6 space end cell row 1 cell space space space 5 end cell cell space space 2 end cell row cell negative 11 end cell cell negative 1 end cell cell space space 5 end cell end table close square brackets space equals space open square brackets table row 3 cell space space space space 1 end cell cell space space space space space minus 11 end cell row cell negative 12 end cell cell space space space space 5 end cell cell space space space space minus 1 end cell row 6 cell space space space space 2 end cell cell space space space space space space space 5 end cell end table close square brackets$

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