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If $straight A space equals open square brackets table row 2 cell space space space space space space space 3 end cell row 1 cell space space minus 4 end cell end table close square brackets comma space straight B space equals space open square brackets table row cell space space space 1 end cell cell space space space space minus 2 end cell row cell negative 1 end cell cell space space space space space space 3 end cell end table close square brackets comma$
verify that (AB)^–1 = B^–1 A^–1

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192.

Let $straight A space equals space open square brackets table row 3 cell space space space 7 end cell row 2 cell space space 5 end cell end table close square brackets space and space straight B space equals space open square brackets table row 6 cell space space 8 end cell row 7 cell space space 9 end cell end table close square brackets comma$ verify that (AB)^–1 = B^–1 A^–1.

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193. Verify that (AB)^–1 = B^–1 A^–1 for the matrices A and B where

straight A space equals space open square brackets table row 3 cell space space 2 end cell row 7 cell space space 5 end cell end table close square brackets comma space space straight B space equals space open square brackets table row 4 cell space space 6 end cell row 3 cell space space 2 end cell end table close square brackets

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194. Verify that (AB)^–1 = B^–1 A^–1 for the matrices A and B where

straight A space equals open square brackets table row 3 cell space space 2 end cell row 7 cell space space 5 end cell end table close square brackets comma space space straight B space equals space open square brackets table row 6 cell space space space space 7 end cell row 8 cell space space space space 9 end cell end table close square brackets

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195. Verify that (AB)^–1 = B^–1 A^–1 for the matrices A and B where

straight A space equals space open square brackets table row 2 cell space space space 1 end cell row 5 cell space space space 3 end cell end table close square brackets comma space space space straight B space equals space open square brackets table row 4 cell space space space 5 end cell row 3 cell space space 4 end cell end table close square brackets

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Short Answer Type

196.
Find (AB)^–1 if $straight A space equals space open square brackets table row 3 cell space 4 end cell row 1 cell space 1 end cell end table close square brackets comma space space straight B to the power of negative 1 end exponent space equals space open square brackets table row 4 cell space space 3 end cell row 2 cell space space 1 end cell end table close square brackets$

Here space straight A space equals open square brackets table row 3 cell space space 4 end cell row 1 cell space space 1 end cell end table close square brackets comma space space straight B to the power of negative 1 end exponent space equals space open square brackets table row 4 cell space space space 3 end cell row 2 cell space space 1 end cell end table close square brackets space space space space space space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 3 cell space space space 4 end cell row 1 cell space space space 1 end cell end table close vertical bar space equals space 3 minus 4 space space space space space minus 1 space not equal to space 0 rightwards double arrow space space space space space straight A to the power of negative 1 end exponent space exists. space

Co-factors of elements of first row of | A | are 1, – 1 respectively.
Co-factors of elements of second row of | A | are –4, 3 respectively.

therefore space space space space adj. space straight A space equals space open square brackets table row 1 cell space space minus 1 end cell row cell negative 4 end cell cell space space space 3 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell space space 1 end cell cell space space space space minus 4 end cell row cell negative 1 end cell cell space space space space space space 3 end cell end table close square brackets space space space space space space space space space straight A to the power of negative 1 end exponent space space space equals space fraction numerator adj space straight A over denominator open vertical bar straight A close vertical bar end fraction space equals space minus open square brackets table row cell space space space 1 end cell cell space space space minus 4 end cell row cell negative 1 end cell cell space space space space space space 3 end cell end table close square brackets therefore space space space space left parenthesis AB right parenthesis to the power of negative 1 end exponent 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 minus open square brackets table row 1 cell space space space 2 end cell row 1 cell space space space 4 end cell end table close square brackets space open square brackets table row cell space space 1 end cell cell space space space minus 4 end cell row cell negative 1 end cell cell space space space space space 3 end cell end table close square brackets space equals space minus open square brackets table row cell 1 minus 2 end cell cell space space space space minus 4 plus 6 end cell row cell 1 minus 4 end cell cell space space space space minus 4 plus 12 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 equals negative open square brackets table row cell negative 1 end cell cell space space space 2 end cell row cell negative 3 end cell cell space space space 8 end cell end table close square brackets space equals space open square brackets table row 1 cell space space minus 2 end cell row 3 cell space space minus 8 end cell end table close square brackets

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197.

Find (AB)^–1 if $straight A space equals space open square brackets table row 5 cell space space 0 end cell row 2 cell space space space 3 end cell end table close square brackets comma space space space straight B to the power of negative 1 end exponent space equals space open square brackets table row 1 cell space space space 2 end cell row 1 cell space space 4 end cell end table close square brackets$

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Long Answer Type

198.

Find the inverse of the matrix
$straight A space equals space open square brackets table row cell negative 1 end cell cell space space 1 end cell cell space space 2 end cell row cell space space space 3 end cell cell negative 1 end cell cell space space 1 space end cell row cell space minus 1 end cell cell space space 3 end cell cell space 4 end cell end table close square brackets.$

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199. Find the inverse 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 0 cell space space space space space space 2 end cell cell space space space minus 3 end cell row 3 cell space space space minus 2 end cell cell space space space space space space 4 end cell end table close square brackets.

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200.

If $straight A space equals space open square brackets table row 0 cell space space space 0 end cell cell space space 1 end cell row 0 cell space space 1 end cell cell space space 0 end cell row 1 cell space space 0 end cell cell space space 0 end cell end table close square brackets comma$ show that A^–1 = A.

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196. Find (AB)–1 if

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