If A=`[(3 , -4), (1 , -1) ]` , then prove that `A^n=[(1+2n , -4n), (n , 1-2n) ]` , where n is any positive integer.

Answer

N/a

Explanation

For n=1
`a^(n) =[{:(1+2n,-4n),(n,1-2n):}],=[{:(3,-4),(1,-1):}]` which is true
therefore ,`A^(n) ` is any postitive integer .
`therefore a^(n) =[{:(1+2n,-4k),(k,1-2k):}] `
for n=K+1,
`A^(K+1)=A^(k).A `
`=[{:(1+2k,-4k),(k,1-2k):}][{:(3,-4),(1,-1):}]`
`=[{:(3+6k-4k,-4-8k+4k),(3k+1-2k,-4k-1k+2k):}]`
`=[{:(1+2(k+1),-4(k+1)),(1+k,1-2(k+1)):}]`
therefore ,`A^(n)` is also true for `n=K+1.`
thus , by the priciple of mathematical inducation , `A^(n)` is true all positive integers n. henceproved.