Find the inverse of the matrix $open square brackets table row 1 cell space space 2 end cell cell space space 3 end cell row 0 cell space space 2 end cell cell space space 4 end cell row 0 cell space space 0 end cell cell space space 5 end cell end table close square brackets$ .

Solution

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 0 cell space space 2 end cell cell space space 4 end cell row 0 cell space 0 end cell cell space space 5 end cell end table close square brackets

therefore space space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space 2 end cell cell space space 3 end cell row 0 cell space space 2 end cell cell space space 4 end cell row 0 0 cell space space 5 end cell end table close vertical bar space equals space open vertical bar table row 2 cell space space space 4 end cell row 0 cell space space space 5 end cell end table close vertical bar space equals space 10 minus 0 space equals space 10 space not equal to space 0 space space space space rightwards double arrow space space space straight A to the power of negative 1 end exponent space exists.

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

open vertical bar table row 2 cell space space 4 end cell row 0 cell space space 5 end cell end table close vertical bar comma space space space space minus open vertical bar table row 0 cell space space 4 end cell row 0 cell space space 5 end cell end table close vertical bar comma space space space space open vertical bar table row 0 cell space space space 2 end cell row 0 cell space space space 0 end cell end table close vertical bar

i.e. 10, 0, 0 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 space 3 end cell row 0 cell space space space space 5 end cell end table close vertical bar comma space space space space open vertical bar table row 1 cell space space space space space 3 end cell row 0 cell space space space space space 5 end cell end table close vertical bar comma space space minus open vertical bar table row 1 cell space space space space 2 end cell row 0 cell space space space space 0 end cell end table close vertical bar space

i.e. – 10, 5, 0 respectively.
Co-factors of the elements of the third row of | A | are

open vertical bar table row 2 cell space space space 3 end cell row 2 cell space space space 4 end cell end table close vertical bar comma space space space space minus open vertical bar table row 1 cell space space space space 3 end cell row 0 cell space space space space 4 end cell end table close vertical bar comma space space space open vertical bar table row 1 cell space space space space 2 end cell row 0 cell space space space space 2 end cell end table close vertical bar

i.e., 2, – 4, 2 respectively.

therefore space space space adj. space straight A space equals space open square brackets table row 10 cell space space space space space 0 end cell cell space space space space space 0 end cell row cell negative 10 end cell cell space space space space space 5 end cell cell space space space space space 0 end cell row 2 cell space space minus 4 end cell cell space space space space space 2 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 10 cell space space minus 10 end cell cell space space space 2 end cell row 0 cell space space space 5 end cell cell negative 4 end cell row 0 cell space 0 end cell cell space 2 end cell end table close square brackets. Now space space straight A to the power of negative 1 end exponent 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 1 over 10 open square brackets table row 10 cell space space minus 10 end cell cell space space space space space space 2 end cell row 0 cell space space space space 5 end cell cell space space minus 4 end cell row 0 cell space space space 0 end cell cell space space space 2 end cell end table close square brackets

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