Question

If A = $open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets space and space straight B equals open square brackets table row 0 1 row 1 0 end table close square brackets$ then show that AB ≠ BA.

Solution

A equals open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets space and space straight B equals open square brackets table row 0 1 row 1 0 end table close square brackets AB equals open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets space open square brackets table row 0 1 row 1 0 end table close square brackets equals open square brackets table row cell 0 plus 0 end cell cell 1 plus 0 end cell row cell 0 minus 1 end cell cell 0 plus 0 end cell end table close square brackets equals open square brackets table row 0 1 row cell negative 1 end cell 0 end table close square brackets BA equals open square brackets table row 0 1 row 1 0 end table close square brackets space open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets equals open square brackets table row cell 0 plus 0 end cell cell 0 minus 1 end cell row cell 1 plus 0 end cell cell 0 minus 0 end cell end table close square brackets equals open square brackets table row 0 1 row cell negative 1 end cell 0 end table close square brackets therefore space space space AB space not equal to space BA space

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