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Determinants

Short Answer Type

161.

Compute the adjoint of the matrix $straight A space equals space open square brackets table row 1 cell space space space space space space 2 end cell row 3 cell space space minus 5 end cell end table close square brackets$
and verify that A (Adj A) = | A | 1.

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Long Answer Type

162.

Verify that A (adj A) = (adj A) = | A | I
where $straight A space equals space open square brackets table row 2 cell space space space 1 end cell cell space space space space space 5 end cell row 3 cell space minus 2 end cell cell space space space minus 4 end cell row cell negative 3 end cell cell space 1 end cell cell space space minus 2 end cell end table close square brackets$

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163.

Verify that A (adj A) = (adj A) = | A | I
where

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Short Answer Type

164. Verify A(adj A) = (adj A) A = | A | 1 for

straight A space equals space open square brackets table row cell space space space 2 end cell cell space space space space 3 end cell row cell negative 4 end cell cell space space minus 6 end cell end table close square brackets.

69 Views

Long Answer Type

165.

Verify A(adj A) = (adj A) A = | A | 1 for the matrix
$straight A space equals space open square brackets table row 1 cell space space space space 1 end cell cell space space space space space space 2 end cell row 3 cell space space space space 0 end cell cell space space minus 2 end cell row 1 cell space space space space 0 end cell cell space space space space space 3 end cell end table close square brackets.$

73 Views

166.
Verify that A(adj. A) = (adj. A) A = | A | I where
$straight A space equals space open square brackets table row 1 cell space space 2 end cell cell space space 3 end cell row 2 cell space space 3 end cell cell space space 2 end cell row 3 cell space 3 end cell cell space space 4 end cell end table close square brackets$

Here space straight A space equals space open square brackets table row 1 cell space space space 2 end cell cell space space space 3 end cell row 2 cell space space space space 3 end cell cell space space space space 2 end cell row 3 cell space space space space 3 end cell cell space space space 4 end cell end table close square brackets therefore space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space space 2 end cell cell space space 3 end cell row 2 cell space space 3 end cell cell space 2 end cell row 3 cell space space 3 end cell cell space space 4 end cell end table close vertical bar space equals space 1 space open vertical bar table row 3 cell space space space space 2 end cell row 3 cell space space space space 4 end cell end table close vertical bar minus 2 space open vertical bar table row 2 cell space space space 2 end cell row 3 cell space space space 4 end cell end table close vertical bar space plus space 3 space open vertical bar table row 2 cell space space space space 3 end cell row 3 cell 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 1 left parenthesis 12 minus 6 right parenthesis minus 2 left parenthesis 8 minus 6 right parenthesis plus 3 left parenthesis 6 minus 9 right parenthesis space equals space 6 minus 4 minus 9 space equals space minus 7

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

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

i.e. 6, – 2, – 3 respectively
Co-factors of the elements of the second row of | A | are

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

i.e. 1, – 5, 3 respectively.
Co-factors of the elements of the third row of | A | are

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

i.e., – 5, 4, – 1 respectively

therefore space space space adj. space straight A space equals space open square brackets table row 6 cell space space space minus 2 end cell cell space space space minus 3 end cell row 1 cell space space minus 5 end cell cell space space space space space 3 end cell row cell negative 5 end cell cell space space space 4 end cell cell space space space space 1 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 6 cell space space space space space 1 space end cell cell space space minus 5 end cell row cell negative 2 end cell cell space space minus 5 end cell cell space space space space space space 4 end cell row cell negative 3 end cell cell space space space 3 end cell cell space space minus 1 end cell end table close square brackets straight A left parenthesis adj. space straight A right parenthesis space equals space open square brackets table row 1 cell space space 2 end cell cell space space 3 end cell row 2 cell space 3 end cell cell space space 2 end cell row 3 cell space 3 end cell cell space space 4 end cell end table close square brackets space open square brackets table row 6 cell space space space 1 end cell cell space space minus 5 end cell row cell negative 2 end cell cell space space minus 5 end cell cell space space space space space space 4 end cell row cell negative 3 end cell cell space space 3 end cell cell space space minus 1 end cell end table close square brackets space space space space space space space space space space space space space space space space equals space open square brackets table row cell 6 minus 4 minus 9 end cell cell space space space 1 minus 10 plus 9 end cell cell space space space space space space minus 5 plus 8 minus 3 end cell row cell 12 minus 6 minus 6 end cell cell space space 2 minus 15 plus 6 end cell cell space space space space space minus 10 plus 12 minus 2 end cell row cell 18 minus 6 minus 12 end cell cell space space space 3 minus 5 plus 12 end cell cell space space space space space space space minus 15 plus 12 minus 4 end cell end table close square brackets space equals space open square brackets table row cell negative 7 end cell 0 0 row 0 cell negative 7 end cell 0 row 0 0 cell negative 7 end cell end table close square brackets space space space space space space space space space space space space space space space space equals negative 7 open square brackets 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 square brackets space equals space minus 7 space straight I therefore space space space space space straight A left parenthesis adj. space straight A right parenthesis space equals space open vertical bar straight A close vertical bar space straight I space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Again space left parenthesis adj. space straight A right parenthesis thin space straight A space equals space open square brackets table row 6 cell space space 1 end cell cell space space minus 5 end cell row cell negative 2 end cell cell space space minus 5 end cell cell space space space 4 end cell row cell negative 3 end cell cell space space 3 end cell cell space minus 1 end cell end table close square brackets space open square brackets table row 1 cell space space space 2 end cell cell space space 3 end cell row 2 cell space 3 end cell cell space space space 2 end cell row 3 cell space space 3 end cell cell space 4 end cell end table close square brackets space space space space space equals space open square brackets table row cell 6 plus 2 minus 15 end cell cell space space space 12 plus 3 minus 15 end cell cell space space space space space 18 plus 2 minus 20 end cell row cell negative 2 minus 10 plus 12 end cell cell space space space minus 4 minus 15 plus 12 end cell cell space space space space space minus 6 minus 10 plus 16 end cell row cell negative 3 plus 6 minus 3 end cell cell negative 6 plus 9 minus 3 end cell cell negative 9 plus 6 minus 4 end cell end table close square brackets space equals space open square brackets table row cell negative 7 end cell cell space space 0 end cell cell space space space 0 end cell row 0 cell negative 7 end cell cell space space space 0 end cell row 0 0 cell negative 7 end cell end table close square brackets space space space equals space minus 7 open square brackets 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 space 0 end cell cell space space space 1 end cell end table close square brackets space equals space minus 7 straight I therefore space space space left parenthesis adj. space straight A right parenthesis thin space straight A space equals space open vertical bar straight A close vertical bar space straight I space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

From (1) and (2), A (adj. A) = (adj. A) A = | A | I

98 Views

167.

If $straight A space equals space open square brackets table row 1 cell space minus 1 end cell cell space space 1 end cell row 2 cell space 3 end cell cell space space 0 end cell row 18 cell space 2 end cell cell space space 10 end cell end table close square brackets comma$
then verify that A (adj. A) = O.

74 Views

Short Answer Type

168.

Find the inverse of $open square brackets table row 2 cell space space space minus 2 end cell row 4 cell space space space space space space 3 end cell end table close square brackets$

71 Views

169.

Find the inverse of $open square brackets table row cell negative 1 end cell cell space space space space 5 end cell row cell negative 3 end cell cell space space space space 2 end cell end table close square brackets$