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Let A=[(2,1),(0,3)] be a matrix. If A^(1...

Let `A=[(2,1),(0,3)]` be a matrix. If `A^(10)=[(a,b),(c,d)]` then prove that `a+d` is divisible by 13.

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We have
`A^(2)=[(2,1),(0,3)][(2,1),(0,3)]=[(4,5),(0,9)]`
`A^(3)=A^(2)A=[(4,5),(0,9)][(2,1),(0,3)]=[(8,19),(0,27)]`
`implies A^(n) =[(2^(n),3^(n)-2^(n)),(0, 3^(n))]`
Now `A^(10)=[(a,b),(c,d)]`
`implies a=2^(10), d=3^(10)`
So, `a+b=2^(10)+3^(10)=4^(5)+9^(5)`, which is multiple of 13.
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