Question

$if space straight A equals open square brackets table row 1 1 1 row 2 0 cell negative 3 end cell row 3 cell negative 1 end cell 2 end table close square brackets comma straight B equals open square brackets table row 1 3 row 0 2 row cell negative 1 end cell 4 end table close square brackets comma space and space straight C equals open square brackets table row 1 2 3 cell negative 4 end cell row 2 0 cell negative 2 end cell 1 end table close square brackets$

Find A(BC), (AB)C and show that (AB(BC).)C=A(BC)

Solution

straight A equals open square brackets table row 1 1 cell negative 1 end cell row 2 0 3 row 3 cell negative 1 end cell 2 end table close square brackets comma straight B equals open square brackets table row 1 3 row 0 2 row cell negative 1 end cell 4 end table close square brackets comma space a n d space straight C equals open square brackets table row 1 2 3 cell negative 4 end cell row 2 0 cell negative 2 end cell 1 end table close square brackets AB equals open square brackets table row 1 1 cell negative 1 end cell row 2 0 3 row 3 cell negative 1 end cell 2 end table close square brackets space open square brackets table row 1 3 row 0 2 row cell negative 1 end cell 4 end table close square brackets space equals open square brackets table row cell 1 plus 0 plus 1 end cell cell 3 plus 2 minus 4 end cell row cell 2 plus 0 minus 3 end cell cell 6 plus 0 plus 12 end cell row cell 3 plus 0 minus 2 end cell cell 9 minus 2 plus 8 end cell end table close square brackets space equals space open square brackets table row 2 1 row cell negative 1 end cell 18 row 1 15 end table close square brackets space AB left parenthesis straight C right parenthesis equals open square brackets table row 2 1 row cell negative 1 end cell 18 row 1 15 end table close square brackets space open square brackets table row 1 2 3 cell negative 4 end cell row 2 0 cell negative 2 end cell 1 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 2 plus 2 end cell cell 4 plus 0 end cell cell 6 minus 2 end cell cell negative 8 plus 1 end cell row cell negative 1 plus 36 end cell cell negative 2 plus 0 end cell cell negative 3 minus 36 end cell cell 4 plus 18 end cell row cell 1 plus 30 end cell cell 2 plus 0 end cell cell 3 minus 36 end cell cell negative 4 plus 15 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 4 4 4 cell negative 7 end cell row 35 cell negative 2 end cell cell negative 39 end cell 22 row 31 2 cell negative 27 end cell 11 end table close square brackets NOW BC space equals space open square brackets table row 1 3 row 0 2 row cell negative 1 end cell 4 end table close square brackets space open square brackets table row 1 2 3 cell negative 4 end cell row 2 0 cell negative 2 end cell 1 end table close square brackets space space space space space space space space equals open square brackets table row cell 1 plus 6 end cell cell 2 plus 0 end cell cell 3 minus 6 end cell cell negative 4 plus 3 end cell row cell 0 plus 4 end cell cell 0 plus 0 end cell cell 0 minus 4 end cell cell 0 plus 2 end cell row cell negative 1 plus 8 end cell cell negative 2 plus 0 end cell cell negative 3 minus 8 end cell cell 4 plus 4 end cell end table close square brackets space space space space space space space space space equals space open square brackets table row 7 2 cell negative 3 end cell cell negative 1 end cell row 4 0 cell negative 4 end cell 2 row 7 cell negative 2 end cell cell negative 11 end cell 8 end table close square brackets space space

therefore space straight A left parenthesis BC right parenthesis equals open square brackets table row 1 1 cell negative 1 end cell row 2 0 3 row 3 cell negative 1 end cell 2 end table close square brackets space open square brackets table row 7 2 cell negative 3 end cell cell negative 1 end cell row 4 0 cell negative 4 end cell 2 row 7 cell negative 2 end cell cell negative 11 end cell 8 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 7 plus 4 minus 7 end cell cell 2 plus 0 plus 2 end cell cell negative 3 minus 4 plus 11 end cell cell negative 1 plus 2 minus 8 end cell row cell 14 plus 0 plus 21 end cell cell 4 plus 0 minus 6 end cell cell negative 6 plus 0 minus 33 end cell cell negative 2 plus 0 plus 24 end cell row cell 21 minus 4 plus 14 end cell cell 6 plus 0 minus 4 end cell cell negative 9 plus 4 minus 22 end cell cell negative 3 minus 2 plus 16 end cell end table close square brackets space space space space space space space space space space space space equals space open square brackets table row 4 4 4 cell negative 7 end cell row 35 cell negative 2 end cell cell negative 39 end cell 22 row 31 2 cell negative 27 end cell 11 end table close square brackets space space space space space space space space therefore space left parenthesis AB right parenthesis straight C space equals space straight A left parenthesis BC right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space

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