If A = $open square brackets table row 4 2 row cell negative 1 end cell 1 end table close square brackets space and space straight I equals open square brackets table row 1 0 row 0 1 end table close square brackets$ show that (A-2I)(A-3I)=O.

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Solution

straight A equals open square brackets table row 4 2 row cell negative 1 end cell 1 end table close square brackets. space straight I equals open square brackets table row 1 0 row 0 1 end table close square brackets straight A minus 2 space straight I equals open square brackets table row 4 2 row cell negative 1 end cell 1 end table close square brackets minus 2 space open square brackets table row 1 0 row 0 1 end table close square brackets equals open square brackets table row 4 2 row cell negative 1 end cell 1 end table close square brackets minus open square brackets table row 2 0 row 0 2 end table close square brackets space space space space space space space space space equals open square brackets table row cell 4 minus 2 end cell cell 2 minus 0 end cell row cell negative 1 minus 0 end cell cell 0 minus 2 end cell end table close square brackets equals open square brackets table row 2 2 row cell negative 1 end cell cell negative 1 end cell end table close square brackets straight A minus 3 space straight I space equals space open square brackets table row 4 2 row cell negative 1 end cell 1 end table close square brackets minus 3 open square brackets table row 1 0 row 0 1 end table close square brackets equals open square brackets table row cell 4 minus 3 end cell cell 2 minus 0 end cell row cell negative 1 minus 0 end cell cell 1 minus 3 end cell end table close square brackets space space space space straight L. straight H. straight S. space equals space left parenthesis straight A minus 2 space straight I right parenthesis left parenthesis straight A minus 3 space straight I right parenthesis space space space space space space space space space equals space open square brackets table row 2 2 row cell negative 1 end cell cell negative 1 end cell end table close square brackets open square brackets table row 1 2 row cell negative 1 end cell cell negative 2 end cell end table close square brackets equals open square brackets table row cell 2 minus 2 end cell cell 4 minus 4 end cell row cell negative 1 plus 1 end cell cell negative 2 plus 2 end cell end table close square brackets space space space space space space space space space equals space open square brackets table row 0 0 row 0 0 end table close square brackets equals 0 equals straight R. straight H. straight S.

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Matrices

If A = $open square brackets table row 4 2 row cell negative 1 end cell 1 end table close square brackets space and space straight I equals open square brackets table row 1 0 row 0 1 end table close square brackets$ show that (A-2I)(A-3I)=O.

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