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If A and P are the square matrices of th...

If A and P are the square matrices of the same order and if P be invertible, show that the matrices A and `P^(-1)` have the same characteristic roots.

Text Solution

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Let `P^(-1)AP=B`
`therefore " " |B-lambdaI|=|P^(-1)AP-lambdaI\|`
`=|P^(-1)AP-P^(-1)lambdaP|`
`=|P^(-1)(A-lambdaI)p|`
`=P^(-1)||A-lambdaI||P|`
`=(1)/(|P|)|A-lambda||P|=|A-lambdaI|`
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