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If A=[(3,-4),(1,-1)] then find tr. (A^(2...

If `A=[(3,-4),(1,-1)]` then find tr. `(A^(2012))`.

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`A=[(3,-4),(1,-1)]=[(1,0),(0,1)]+[(2,-4),(1,-2)]=I+B`
Now, `B^(2)=[(2,-4),(1,-2)][(2,-4),(1,-2)]=[(0,0),(0,0)]=O`
`implies A^(2012)=(I+B)^(2012)`
`=I^(2012)+^(2012)C_(1)I^(2012) B+O+O+..." "( :' B^(2)=B^(3)=B^(4) ... O)`
`=I+2012 B`
`=[(1,0),(0,1)]+2012 [(2,-4),(1,-2)]`
`:.` Trace of `A^(2012)=2+2012 (2-2)`
`=2`
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