Let the matrix A = the first line 1 1 1, the second line 2 2 2, the third line - 1 - 1 - 1, the fourth line 3 3 3 3 find the nth power of A | Math enthusiast | Mathematical art

Let the matrix A = the first line 1 1 1, the second line 2 2 2, the third line - 1 - 1 - 1, the fourth line 3 3 3 3 find the nth power of A

This is the case where the rank of a is 1
Let α = (1,2, - 1,3) ^ t, β = (1,1,1,1) ^ t
Then a = α β ^ t
Because β ^ t α = 1 + 2-1 + 3 = 5
So a ^ n = (α β ^ t) ^ n
= (αβ^T)(αβ^T)(αβ^T)...(αβ^T)(αβ^T)
= α(β^Tα)(β^Tα)β^T...α(β^Tα)β^T
= (β^Tα)^(n-1)(αβ^T)
= 5^(n-1)A.

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