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Problem: Is Periodic matrix [[1,-2,-6],[-3,2,9],[2,0,-3]] [ Calculator, Method and examples ]

Solution:
Your problem -> Is Periodic matrix [[1,-2,-6],[-3,2,9],[2,0,-3]]

A square matrix A is called a periodic matrix, if A^m = A for some positive integer m.

A =
 1 -2 -6 -3 2 9 2 0 -3

A×A=
 1 -2 -6 -3 2 9 2 0 -3
×
 1 -2 -6 -3 2 9 2 0 -3

=
 1×1-2×-3-6×2 1×-2-2×2-6×0 1×-6-2×9-6×-3 -3×1+2×-3+9×2 -3×-2+2×2+9×0 -3×-6+2×9+9×-3 2×1+0×-3-3×2 2×-2+0×2-3×0 2×-6+0×9-3×-3

=
 1+6-12 -2-4+0 -6-18+18 -3-6+18 6+4+0 18+18-27 2+0-6 -4+0+0 -12+0+9

=
 -5 -6 -6 9 10 9 -4 -4 -3

(A^2)×A=
 -5 -6 -6 9 10 9 -4 -4 -3
×
 1 -2 -6 -3 2 9 2 0 -3

=
 -5×1-6×-3-6×2 -5×-2-6×2-6×0 -5×-6-6×9-6×-3 9×1+10×-3+9×2 9×-2+10×2+9×0 9×-6+10×9+9×-3 -4×1-4×-3-3×2 -4×-2-4×2-3×0 -4×-6-4×9-3×-3

=
 -5+18-12 10-12+0 30-54+18 9-30+18 -18+20+0 -54+90-27 -4+12-6 8-8+0 24-36+9

=
 1 -2 -6 -3 2 9 2 0 -3

Here A^3 = A, so A is a periodic matrix of period 2

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