Question

$Given space that space straight A equals space open square brackets table row cell negative 4 end cell cell space space space 4 end cell cell space space 4 end cell row cell negative 7 end cell cell space space 1 end cell cell space space space 3 end cell row 5 cell space space minus 3 end cell cell space space minus 1 end cell end table close square brackets space space and space space straight B space equals space open square brackets table row 1 cell space space minus 1 end cell cell space space 1 end cell row 1 cell space space minus 2 end cell cell space space minus 2 end cell row cell negative 2 end cell cell space space 1 end cell cell space space space space 3 end cell end table close square brackets space Find space AB. space Use space this space to space solve space the space following space system space of space linear space equations space colon space space straight x space – space straight y space plus space straight z space equals space 4 comma straight x space – space 2 straight y space – space 2 straight z space equals space 9 comma space 2 space straight x space plus space straight y space plus space 3 space straight z space equals space 1.$

Solution

space straight A equals space open square brackets table row cell negative 4 end cell cell space space space 4 end cell cell space space 4 end cell row cell negative 7 end cell cell space space 1 end cell cell space space space 3 end cell row 5 cell space space minus 3 end cell cell space space minus 1 end cell end table close square brackets space space comma space straight B space equals space open square brackets table row 1 cell space space minus 1 end cell cell space space 1 end cell row 1 cell space space minus 2 end cell cell space space minus 2 end cell row cell negative 2 end cell cell space space 1 end cell cell space space space space 3 end cell end table close square brackets AB equals space open square brackets table row cell negative 4 end cell cell space space space 4 end cell cell space space 4 end cell row cell negative 7 end cell cell space space 1 end cell cell space space space 3 end cell row 5 cell space space minus 3 end cell cell space space minus 1 end cell end table close square brackets space space open square brackets table row 1 cell space space minus 1 end cell cell space space 1 end cell row 1 cell space space minus 2 end cell cell space space minus 2 end cell row cell negative 2 end cell cell space space 1 end cell cell space space space space 3 end cell end table close square brackets space space space space space space equals open square brackets table row cell negative 4 plus 4 plus 8 end cell cell space space space space space 4 minus 8 plus 4 end cell cell space space minus 4 minus 8 plus 12 end cell row cell negative 7 plus 1 plus 6 end cell cell space space space space 7 minus 2 plus 3 end cell cell space space space space minus 7 minus 2 plus 9 end cell row cell 5 minus 3 minus 2 end cell cell space space minus 5 plus 6 minus 1 end cell cell space space space space 5 plus 6 minus 3 end cell end table close square brackets therefore space AB equals space open square brackets table row 8 cell space space 0 end cell cell space 0 end cell row 0 cell space space 8 end cell cell space 0 end cell row 0 cell space space 0 end cell cell space 8 end cell end table close square brackets Consider space the space equations space space straight x space – space straight y space plus space straight z space equals space 4 space space straight x space – space 2 straight y space – space 2 straight z space equals 9 space space 2 straight x space plus space straight y space plus space 3 straight z space equals space straight l space These space equations space can space be space written space as space space open square brackets table row 1 cell space minus 1 end cell 1 row 1 cell space space minus 2 end cell cell space space minus 2 end cell row 2 cell space space space 1 end cell 3 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 equals space open square brackets table row 4 row 9 row 1 end table close square brackets Premultiplying space both space sides space by space space space open square brackets table row cell negative 4 end cell cell space space space 4 end cell cell space space 4 end cell row cell negative 7 end cell cell space space 1 end cell cell space space space 3 end cell row 5 cell space space minus 3 end cell cell space space minus 1 end cell end table close square brackets space space space we space get comma space open square brackets table row cell negative 4 end cell cell space space space 4 end cell cell space space 4 end cell row cell negative 7 end cell cell space space 1 end cell cell space space space 3 end cell row 5 cell space space minus 3 end cell cell space space minus 1 end cell end table close square brackets space space open square brackets table row 1 cell space space minus 1 end cell cell space space 1 end cell row 1 cell space space minus 2 end cell cell space space minus 2 end cell row cell negative 2 end cell cell space space 1 end cell cell space space space space 3 end cell 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 4 end cell cell space space space 4 end cell cell space space 4 end cell row cell negative 7 end cell cell space space 1 end cell cell space space space 3 end cell row 5 cell space space minus 3 end cell cell space space minus 1 end cell end table close square brackets space space open square brackets table row 4 row 9 row 1 end table close square brackets or space space space space space open square brackets table row 8 cell space space 0 end cell cell space 0 end cell row 0 cell space space 8 end cell cell space 0 end cell row 0 cell space space 0 end cell cell space 8 end cell 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 space equals space open square brackets table row cell negative 16 plus 36 plus 4 end cell row cell negative 28 plus 9 plus 3 end cell row cell 20 minus 27 minus 1 end cell end table close square brackets therefore space space open square brackets table row cell 8 straight x end cell row cell 8 straight y end cell row cell 8 straight z end cell end table close square brackets space equals space open square brackets table row 24 row cell negative 16 end cell row cell negative 8 end cell end table close square brackets therefore space space 8 straight x equals 24 space comma space 8 straight y equals negative 16. space 8 straight z equals negative 8 therefore space space straight x equals 3 straight y equals negative 2 space straight z equals negative 1

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