Question

$If space straight A space equals space open vertical bar table row 2 cell space space 0 end cell cell negative 1 end cell row 5 cell space 1 end cell cell space 0 end cell row 0 cell space 1 end cell cell space 3 end cell end table close vertical bar comma space then space find thin space straight A to the power of negative 1 end exponent space using space elementary space row space operations.$ $Applying space straight R subscript 3 space rightwards arrow space straight R subscript 3 plus left parenthesis negative 1 right parenthesis straight R subscript 2 straight A to the power of negative 1 end exponent open square brackets table row 1 cell space 0 end cell cell negative 1 half end cell row 0 cell space 1 end cell cell space 5 over 2 end cell row 0 cell space 1 end cell cell 1 half end cell end table close square brackets space equals space open square brackets table row cell 1 half end cell cell space space 0 end cell cell space 0 end cell row cell negative 5 over 2 end cell 1 cell space 0 end cell row cell 5 over 2 end cell cell negative 1 end cell cell space 1 end cell end table close square brackets Applying space straight R subscript 3 space rightwards arrow space left parenthesis 2 right parenthesis straight R subscript 3 straight A to the power of negative 1 end exponent open square brackets table row 1 cell space 0 end cell cell negative 1 half end cell row 0 1 cell space 5 over 2 end cell row 0 cell space 0 end cell 1 end table close square brackets space equals space open square brackets table row cell 1 half end cell 0 cell space space space 0 end cell row cell negative 5 over 2 end cell 1 cell space space 0 end cell row 5 cell negative 2 end cell cell space space 2 end cell end table close square brackets$

Solution

open vertical bar straight A close vertical bar space equals space open vertical bar table row 2 cell space space space 0 end cell cell space minus 1 end cell row 5 cell space space space 1 end cell cell space space space 0 end cell row 0 cell space space space 1 end cell cell space space space 3 end cell end table close vertical bar space space equals space 2 left parenthesis 3 minus 0 right parenthesis minus 0 left parenthesis 15 minus 0 right parenthesis minus 1 left parenthesis 5 minus 0 right parenthesis space space equals 6 minus 0 minus 5 space space equals space 1 space space not equal to 0 space

Hence space straight A to the power of negative 1 end exponent space exists. straight A to the power of negative 1 end exponent straight A space equals space 1 straight A to the power of negative 1 end exponent open square brackets table row 2 cell space space 0 end cell cell space minus 1 end cell row cell space 5 space end cell cell space 1 end cell cell space space space 0 end cell row 0 cell space 1 end cell cell space space space 3 end cell end table close square brackets space equals space open square brackets table row 1 cell space 0 end cell cell space 0 end cell row 0 cell space 1 end cell cell space 0 end cell row 0 cell space 0 end cell cell space 1 end cell end table close square brackets Applying space straight R subscript 1 space rightwards arrow space open parentheses 1 half close parentheses straight R subscript 1 straight A to the power of negative 1 end exponent open square brackets table row 1 cell space 0 end cell cell negative 1 half end cell row 5 1 cell space space 0 end cell row 0 1 cell space 3 end cell end table close square brackets space equals space open square brackets table row cell 1 half end cell cell space 0 end cell cell space space space 0 end cell row 0 cell space 1 end cell cell space space space 0 end cell row 0 cell space 0 end cell cell space space space 1 end cell end table close square brackets Applying space straight R subscript 2 space rightwards arrow space straight R subscript 2 space plus left parenthesis negative 5 right parenthesis straight R subscript 1 straight A to the power of negative 1 end exponent open square brackets table row 1 cell space 0 end cell cell negative 1 half end cell row 0 1 cell space space 5 over 2 end cell row 0 1 cell space 3 end cell end table close square brackets space equals space open square brackets table row cell 1 half end cell cell space space 0 end cell cell space space space 0 end cell row cell negative 5 over 2 end cell cell space 1 end cell cell space space 0 end cell row 0 cell space 0 end cell cell space space space 1 end cell end table close square brackets

Applying space straight R subscript 3 space rightwards arrow straight R subscript 3 plus left parenthesis negative 1 right parenthesis straight R subscript 2 straight A to the power of negative 1 end exponent open square brackets table row 1 cell space space 0 end cell cell space space minus 1 half end cell row 0 cell space space 1 end cell cell space space space space 5 over 2 end cell row 0 cell space space 0 end cell cell space space space space 1 half end cell end table close square brackets space equals space open square brackets table row cell 1 half end cell cell space 0 end cell cell space 0 end cell row cell negative 5 over 2 end cell 1 cell space 0 end cell row cell 5 over 2 end cell cell negative 1 end cell 1 end table close square brackets Applying space straight R subscript 3 space rightwards arrow left parenthesis 2 right parenthesis straight R subscript 3 straight A to the power of negative 1 end exponent open square brackets table row 1 cell space 0 end cell cell space minus 1 half end cell row 0 cell space 1 end cell cell space space 5 over 2 end cell row 0 cell space 0 end cell cell space 1 end cell end table close square brackets space equals open square brackets table row cell 1 half end cell cell space 0 end cell 0 row cell negative 5 over 2 end cell 1 0 row 5 cell negative 2 end cell cell space 2 end cell end table close square brackets

Applying space straight R subscript 1 space rightwards arrow space straight R subscript 1 space plus space open parentheses 1 half close parentheses straight R subscript 3 space space space space space space space space space space space space straight R subscript 2 space rightwards arrow space straight R subscript 2 space plus space open parentheses negative 5 over 2 close parentheses straight R subscript 3 space space straight A to the power of negative 1 end exponent open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets space equals space open square brackets table row 3 cell space minus 1 end cell cell space space space 1 end cell row cell negative 15 end cell cell space space space 6 end cell cell space minus 5 end cell row 5 cell space minus 2 end cell cell space space space 2 end cell end table close square brackets straight A to the power of negative 1 end exponent space equals space open square brackets table row 3 cell space minus 1 end cell cell space space space space 1 end cell row cell negative 15 end cell cell space space space 6 end cell cell space minus 5 end cell row 5 cell space minus 2 end cell cell space space 2 end cell end table close square brackets

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Matrices

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