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Let A=[(1,2),(-1,3)] .If A^6=kA-205I t...

Let `A=[(1,2),(-1,3)]` .If `A^6=kA-205I` then find the numerical quantity of `k-40`

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`A=[(1,2),(-1,3)]`
`implies A^(2)=[(1,2),(-1,3)][(1,2),(-1,3)]=[(-1,8),(-4,7)]=4A-5I`
`implies A^(3)=A A^(2)=A(4A-5I)`
`=4A^(2)-5A`
`=4(4A-5I)-5A`
`=11 A-20 I`
`A^(6)=A^(3) A^(3)=(11A-20I)^(2)`
`=121A^(2)-440 A+400 I`
`=121 (4A-5I)-440 A+400 I`
`=44A-205 I`
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