Question

$use space product space space open square brackets table row 1 cell negative 1 end cell cell space space space 2 end cell row 0 cell space space space 2 end cell cell space space minus 3 end cell row 3 cell negative 2 end cell cell space space space space 4 end cell end table close square brackets space space open square brackets table row cell negative 2 end cell cell space space space 0 end cell cell space space 1 end cell row 9 cell space space space 2 end cell cell space space space minus 3 end cell row 6 cell space space space 1 end cell cell space space space minus 2 end cell end table close square brackets space to space solve space the space system space of space equations$

Solution

Let space straight A equals space open square brackets table row 1 cell negative 1 end cell 2 row 0 cell space space space 2 end cell cell negative 3 end cell row 3 cell negative 2 end cell 4 end table close square brackets space space comma space straight B equals space open square brackets table row cell negative 2 end cell 0 1 row 9 2 cell negative 3 end cell row 6 1 cell negative 2 end cell end table close square brackets therefore space space space space AB equals space open square brackets table row 1 cell negative 1 end cell 2 row 0 cell space space space 2 end cell cell negative 3 end cell row 3 cell negative 2 end cell 4 end table close square brackets space open square brackets table row cell negative 2 end cell 0 1 row 9 2 cell negative 3 end cell row 6 1 cell negative 2 end cell end table close square brackets space space space space space space space space space space space space space space equals open square brackets table row cell negative 2 minus 9 plus 12 end cell cell 0 minus 2 plus 2 end cell cell 1 plus 3 minus 4 end cell row cell 0 plus 18 minus 18 end cell cell space space space space 0 plus 4 minus 3 space space space end cell cell 0 minus 6 plus 6 end cell row cell negative 6 minus 18 plus 24 end cell cell 0 minus 4 plus 4 end cell cell 3 plus 6 minus 8 end cell end table close square brackets space space equals space open square brackets table row 1 cell space space 0 end cell 0 row 0 cell space space space 1 end cell cell space space 0 end cell row 0 cell space space space 0 end cell 1 end table close square brackets therefore space space AB equals space straight I again space BA equals space space space open square brackets table row cell negative 2 end cell 0 1 row 9 2 cell negative 3 end cell row 6 1 cell negative 2 end cell end table close square brackets space space space open square brackets table row 1 cell negative 1 end cell 2 row 0 cell space space space 2 end cell cell negative 3 end cell row 3 cell negative 2 end cell 4 end table close square brackets space space space space space space space space space space space space space space space space space equals space space open square brackets table row cell negative 2 plus 0 plus 3 end cell cell space space 2 plus 0 minus 2 end cell cell negative 4 plus 0 plus 4 end cell row cell 9 plus 0 minus 9 end cell cell space space space minus 9 plus 4 plus 6 end cell cell space space space space 18 minus 6 minus 12 end cell row cell 6 plus 0 minus 6 end cell cell space space minus 6 plus 2 plus 4 end cell cell 12 minus 13 minus 8 end cell end table close square brackets space space equals space open square brackets table row 1 cell space space 0 end cell 0 row 0 cell space space space 1 end cell cell space space 0 end cell row 0 cell space space space 0 end cell 1 end table close square brackets therefore space space BA equals space straight I therefore space space AB equals BA equals space straight I Consider space the space equations space space straight x space – space straight y space plus space 2 space straight z space equals space 1 space space 2 straight y space – space 3 space straight z space equals 1 space space 3 space straight x space – space 2 space straight y space plus space 4 space straight z space equals space 2 space space These space equations space can space be space written space as open square brackets table row 1 cell negative 1 end cell 2 row 0 cell space space space 2 end cell cell negative 3 end cell row 3 cell negative 2 end cell 4 end table close square brackets space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row 1 row 1 row 2 end table close square brackets Pre space multiplying space both space sides space by space space space space open square brackets table row cell negative 2 end cell cell space space space space space space space 0 end cell cell space space space space space space 1 end cell row 9 cell space space space space space space 2 end cell cell space space space minus 3 end cell row 6 cell space space space space space 1 end cell cell space space space minus 2 end cell end table close square brackets space comma space we space get open square brackets table row cell negative 2 end cell cell space space space space space space space 0 end cell cell space space space space space space 1 end cell row 9 cell space space space space space space 2 end cell cell space space space minus 3 end cell row 6 cell space space space space space 1 end cell cell space space space minus 2 end cell end table close square brackets space space space open square brackets table row 1 cell negative 1 end cell 2 row 0 cell space space space 2 end cell cell negative 3 end cell row 3 cell negative 2 end cell 4 end table close square brackets space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row cell negative 2 end cell cell space space space space space space space 0 end cell cell space space space space space space 1 end cell row 9 cell space space space space space space 2 end cell cell space space space minus 3 end cell row 6 cell space space space space space 1 end cell cell space space space minus 2 end cell end table close square brackets space space open square brackets table row 1 row 1 row 2 end table close square brackets or space space open square brackets table row 1 cell space space 0 end cell 0 row 0 cell space space space 1 end cell cell space space 0 end cell row 0 cell space space space 0 end cell 1 end table close square brackets space space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row cell negative 2 plus 0 plus 2 end cell row cell 9 plus 2 minus 6 end cell row cell 6 plus 1 minus 4 end cell end table close square brackets or space open square brackets table row straight x row straight y row straight z end table close square brackets space equals open square brackets table row 0 row 5 row 3 end table close square brackets therefore space straight x space equals space 0 comma space straight y space equals space 5 comma space straight z space equals space 3 space is space required space solution.

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Matrices

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