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Solution
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Solution provided by AtoZmath.com
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Periodic Matrix checker calculator
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 1. `[[1,-2,-6],[-3,2,9],[2,0,-3]]`
2. `[[2,-2,-6],[-3,2,9],[2,0,-3]]`
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Example1. is Periodic Matrix ?
`[[1,-2,-6],[-3,2,9],[2,0,-3]]`Solution: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` | |
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| `A×A` | = | | `1` | `-2` | `-6` | | | `-3` | `2` | `9` | | | `2` | `0` | `-3` | |
| × | | `1` | `-2` | `-6` | | | `-3` | `2` | `9` | | | `2` | `0` | `-3` | |
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| = | | `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` | |
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| = | | `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` | |
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| = | | `-5` | `-6` | `-6` | | | `9` | `10` | `9` | | | `-4` | `-4` | `-3` | |
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| `(A^2)×A` | = | | `-5` | `-6` | `-6` | | | `9` | `10` | `9` | | | `-4` | `-4` | `-3` | |
| × | | `1` | `-2` | `-6` | | | `-3` | `2` | `9` | | | `2` | `0` | `-3` | |
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| = | | `-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` | |
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| = | | `-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` | |
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| = | | `1` | `-2` | `-6` | | | `-3` | `2` | `9` | | | `2` | `0` | `-3` | |
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Here `A^3 = A`, so `A` is a periodic matrix of period 2
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