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A = 1 1 0,0 1 0,0 0 1= 1 0 0 ,0 1 0,0 0 1 + 0 1 0,0 0 0,0 0 0 = Identity matrix + B = 1 + B
A = 1 + B ,where B = 0 1 0,0 0 0,0 0 0 also, B^k = 0 for k>=2. That is B^2=B^3=...=0
now, A^2 = 1+2B+B^2 = 1 + 2B + 0 = 1 + 2B lly, A^3 = 1+3B+B^3 = 1+3B+0 = 1+3B . . Hence A^100 = 1 + 100B = 1 0 0,0 1 0,0 0 1 + 0 100 0,0 0 0,0 0 0 A^100 = 1 100 0, 0 1 0, 0 0 1 = A B 0, 0 C 0, 0 D E A, B, C, D, E = 1, 100, 1, 0 ,1
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Linear Algebra Foundations #5 - The 100th Power of a Matrix
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A = 1 1 0,0 1 0,0 0 1= 1 0 0 ,0 1 0,0 0 1 + 0 1 0,0 0 0,0 0 0
= Identity matrix + B = 1 + B
A = 1 + B ,where B = 0 1 0,0 0 0,0 0 0
also, B^k = 0 for k>=2. That is B^2=B^3=...=0
now, A^2 = 1+2B+B^2 = 1 + 2B + 0 = 1 + 2B
lly, A^3 = 1+3B+B^3 = 1+3B+0 = 1+3B
. . Hence A^100 = 1 + 100B = 1 0 0,0 1 0,0 0 1 + 0 100 0,0 0 0,0 0 0
A^100 = 1 100 0, 0 1 0, 0 0 1 = A B 0, 0 C 0, 0 D E
A, B, C, D, E = 1, 100, 1, 0 ,1