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Find Is Periodic matrix [[1,-2,-6],[-3,2,9],[2,0,-3]]

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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