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Compute (AB)¹ where:
$straight A space equals space open square brackets table row 1 cell space space space space 1 end cell cell space space space 2 end cell row 0 cell space space space 2 end cell cell space minus 3 end cell row 3 cell space minus 2 end cell cell space space space space 4 end cell end table close square brackets space space space space space space space straight B to the power of negative 1 end exponent space equals space open square brackets table row 1 cell space space space space space 2 end cell cell space space space space space 0 end cell row 0 cell space space space 3 end cell cell space space minus 1 end cell row 1 cell space space 0 end cell cell space space space space 2 end cell end table close square brackets$

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222.

If $straight A space equals space open square brackets table row 2 cell space space space 2 end cell cell space space space 1 end cell row cell negative 2 end cell cell space space space 1 end cell cell space space 2 end cell row 1 cell space minus 2 end cell cell space space 2 end cell end table close square brackets space space space and space straight B space equals space open square brackets table row 1 cell space space 3 end cell cell space space 2 end cell row 1 cell space space space 1 end cell cell space 1 end cell row 2 cell negative 3 end cell cell space 1 end cell end table close square brackets comma$ verify that (AB)^–1 = B"^–1 A^–1.

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223.
If $straight A to the power of negative 1 end exponent space equals space open square brackets table row 3 cell space space minus 1 end cell cell space space space space space 1 end cell row cell negative 15 end cell cell space space space space space 6 end cell cell space minus 5 end cell row 5 cell space minus 2 end cell cell space space space space 2 end cell end table close square brackets space space space and space straight B space equals space open square brackets table row cell space space space 1 end cell cell space space space space 2 end cell cell space space space minus 2 end cell row cell negative 1 end cell cell space space space space space 3 end cell cell space space space space space 0 end cell row cell space space 0 end cell cell space space minus 2 end cell cell space space space space space 1 end cell end table close square brackets comma$ find (AB)^–1.

straight A to the power of negative 1 end exponent space equals space open square brackets table row cell space space space 3 end cell cell space space minus 1 end cell cell space space space space space 1 end cell row cell negative 15 end cell cell space space space space space 6 end cell cell space space minus 5 end cell row cell space space 5 end cell cell space space minus 2 end cell cell space space space space 2 end cell end table close square brackets space space comma space straight B space equals open square brackets table row cell space space space 1 end cell cell space space space space 2 end cell cell space space space minus 2 end cell row cell negative 1 end cell cell space space space space 3 end cell cell space space space space space 0 end cell row cell space space space 0 end cell cell space minus 2 end cell cell space space space space 1 end cell end table close square brackets comma

open vertical bar straight B close vertical bar space equals space open vertical bar table row cell space space 1 end cell cell space space space 2 end cell cell space space space minus 2 end cell row cell negative 1 end cell cell space space space space 3 end cell cell space space space space space space 0 end cell row cell space space 0 end cell cell space minus 2 end cell cell space space space space space 1 end cell end table close vertical bar space equals space 1 open vertical bar table row 3 cell space space space 0 end cell row cell negative 2 end cell cell space space space 1 end cell end table close vertical bar minus 2 space open vertical bar table row cell negative 1 end cell cell space space 0 end cell row 0 cell space space 1 end cell end table close vertical bar plus left parenthesis negative 2 right parenthesis space open vertical bar table row cell negative 1 end cell cell space space space space 3 end cell row 0 cell space minus 2 end cell end table close vertical bar space space space space space space space equals 1 left parenthesis 3 minus 0 right parenthesis space minus space 2 left parenthesis negative 1 minus 0 right parenthesis minus 2 left parenthesis 2 minus 0 right parenthesis space equals space 3 plus 2 minus 4 space equals space 1 not equal to space 0 therefore space space space space straight B to the power of negative 1 end exponent space exists. space space space space space space straight B subscript 11 space equals space open vertical bar table row 3 cell space space space 0 end cell row cell negative 2 end cell cell space space 1 end cell end table close vertical bar space equals space 3 minus 0 space equals space 3 space space space space space space straight B subscript 12 space equals space minus open vertical bar table row cell negative 1 end cell cell space space space 0 end cell row cell space 0 end cell cell space space 1 end cell end table close vertical bar space equals space minus left parenthesis negative 1 minus 0 right parenthesis space equals space 1 space space space space space space straight B subscript 13 space equals space open vertical bar table row cell negative 1 end cell cell space space space space 3 end cell row 0 cell space space minus 2 end cell end table close vertical bar space equals space 2 minus 0 space equals space 2 space space space space space straight B subscript 21 space equals space minus open vertical bar table row 2 cell space space space minus 2 end cell row cell negative 2 end cell cell space space space 1 end cell end table close vertical bar space equals space minus left parenthesis 2 minus 4 right parenthesis space equals space 2 space space space space space straight B subscript 22 space equals space open vertical bar table row 1 cell space space space minus 2 end cell row 0 cell space space space space space 1 end cell end table close vertical bar space equals space 1 minus 0 space equals space 1 space space space space straight B subscript 23 space equals space minus open vertical bar table row 1 cell space space space space space space 2 end cell row 0 cell space space minus 2 end cell end table close vertical bar space equals negative left parenthesis negative 2 minus 0 right parenthesis space equals space 2 space space space straight B subscript 31 space equals space open vertical bar table row 2 cell space space space minus 2 end cell row 3 cell space space space space space space 0 end cell end table close vertical bar space equals space 0 plus 6 space equals space 6

straight B subscript 32 space equals space minus open vertical bar table row 1 cell space space space minus 2 end cell row cell negative 1 end cell cell space space space space space space 0 end cell end table close vertical bar space equals space minus left parenthesis 0 minus 2 right parenthesis space equals space 2 straight B subscript 33 space equals space open vertical bar table row cell space space space 1 end cell cell space space space 2 end cell row cell negative 1 end cell cell space space space 3 end cell end table close vertical bar space equals 3 plus 2 space equals 5 space adj. space straight B space equals space open square brackets table row 3 cell space space 1 end cell cell space space 2 end cell row 2 cell space space 1 end cell cell space space 2 end cell row 6 cell space space 2 end cell cell space space 5 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 3 cell space space space space 2 end cell cell space space space 6 end cell row 1 cell space space space 1 end cell cell space space space 2 end cell row 2 cell space space space 2 end cell cell space space space 5 end cell end table close square brackets space therefore space space straight B to the power of negative 1 end exponent space equals space fraction numerator adj. space straight B over denominator open vertical bar straight B close vertical bar end fraction space equals space open square brackets table row 3 cell space space space 2 end cell cell space space 6 end cell row 1 cell space space 1 end cell cell space space 2 end cell row 2 cell space 2 end cell cell space 5 end cell end table close square brackets space space Now space left parenthesis AB right parenthesis to the power of negative 1 end exponent space space equals space straight B to the power of negative 1 end exponent straight A to the power of negative 1 end exponent space equals space open square brackets table row 3 cell space space 2 end cell cell space space 6 end cell row 1 cell space 1 end cell cell space space 2 end cell row 2 cell space 2 end cell cell space space 5 end cell end table close square brackets space space open square brackets table row cell space 3 end cell cell space space minus 1 end cell cell space space 1 end cell row cell negative 15 end cell cell space space space space 6 end cell cell space space minus 5 end cell row 5 cell space space minus 2 end cell cell space space space space 2 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space equals space open square brackets table row cell 9 minus 30 plus 30 end cell cell space space space space space minus 3 plus 12 minus 12 end cell cell space space space space 3 minus 10 plus 12 end cell row cell 3 minus 15 plus 10 end cell cell space minus 1 plus 6 minus 4 end cell cell space 1 minus 5 plus 4 end cell row cell 6 minus 30 plus 25 end cell cell space minus 2 plus 12 minus 10 end cell cell space space space space 2 minus 10 plus 10 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space equals open square brackets table row 9 cell space space minus 3 end cell cell space space space 5 end cell row cell negative 2 end cell cell space space space space 1 end cell cell space space space 0 end cell row 1 cell space space space space 0 end cell cell space space space 2 end cell end table close square brackets

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Multiple Choice Questions

224. Let A be a non-singular square matrix of order 3 × 3. Then | adj A| j is equal to

| A |
| A |²
| A |³
| A |³

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225.

If A is an invertible matrix of order 2, then det (A^–1) is equal to

det (A)
$fraction numerator 1 over denominator det space left parenthesis straight A right parenthesis end fraction$
1
1

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226. If x, y, z are non-zero real numbers, then the inverse of matrix

straight A space equals open square brackets table row straight x cell space space space space 0 end cell cell space space space 0 end cell row 0 cell space space space straight y end cell cell space space space 0 end cell row 0 cell space space space 0 end cell cell space space space straight z end cell end table close square brackets space is

$open square brackets table row cell straight x to the power of negative 1 end exponent end cell cell space space 0 end cell cell space space 0 end cell row 0 cell space space straight y to the power of negative 1 space end exponent end cell cell space space 0 end cell row 0 0 cell space straight z to the power of negative 1 end exponent end cell end table close square brackets$
$xyz open square brackets table row cell straight x to the power of negative 1 end exponent end cell cell space space 0 end cell cell space space space 0 end cell row 0 cell space space straight y to the power of negative 1 end exponent space end cell cell space space 0 end cell row 0 0 cell space straight z to the power of negative 1 end exponent end cell end table close square brackets$
$1 over xyz open square brackets table row straight x cell space space 0 end cell cell space space 0 end cell row 0 cell space straight y end cell cell space space 0 end cell row 0 cell space 0 end cell cell space space straight z end cell end table close square brackets$
$1 over xyz open square brackets table row straight x cell space space 0 end cell cell space space 0 end cell row 0 cell space straight y end cell cell space space 0 end cell row 0 cell space 0 end cell cell space space straight z end cell end table close square brackets$