$If space straight A equals open square brackets table row 1 cell negative 3 end cell 2 row 2 0 2 end table close square brackets comma space straight B equals open square brackets table row 2 cell negative 1 end cell cell negative 1 end cell row 1 0 cell negative 1 end cell end table close square brackets$ .find the matrix C such that A + B + C is a zero matrix.

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Solution

We have

straight A equals open square brackets table row 1 cell negative 3 end cell 2 row 2 0 2 end table close square brackets comma space straight B equals open square brackets table row 2 cell negative 1 end cell cell negative 1 end cell row 1 0 cell negative 1 end cell end table close square brackets

Now A + B + C = O

rightwards double arrow space space space space space straight C minus straight A minus straight B equals negative open square brackets table row 1 cell negative 3 end cell 2 row 2 0 2 end table close square brackets minus open square brackets table row 2 cell negative 1 end cell cell negative 1 end cell row 1 0 cell negative 1 end cell end table close square brackets space space space space space space space space equals open square brackets table row cell negative 1 end cell 3 cell negative 2 end cell row cell negative 2 end cell 0 cell negative 2 end cell end table close square brackets plus open square brackets table row cell negative 2 end cell 1 1 row cell negative 1 end cell 0 1 end table close square brackets equals open square brackets table row cell negative 1 minus 2 end cell cell 3 plus 1 end cell cell negative 2 plus 1 end cell row cell negative 2 minus 1 end cell cell 0 plus 0 end cell cell negative 2 plus 1 end cell end table close square brackets therefore space space space space straight C equals open square brackets table row cell negative 3 end cell 4 cell negative 1 end cell row cell negative 3 end cell 0 cell negative 1 end cell end table close square brackets

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