1. is Nilpotent Matrix ?
`[[1,2,3],[1,2,3],[-1,-2,-3]]`Solution:A square matrix `A` is called a nilpotent matrix, if `A^m = 0` for some positive integer m.
| `A` | = | | `1` | `2` | `3` | | | `1` | `2` | `3` | | | `-1` | `-2` | `-3` | |
|
| `A×A` | = | | `1` | `2` | `3` | | | `1` | `2` | `3` | | | `-1` | `-2` | `-3` | |
| × | | `1` | `2` | `3` | | | `1` | `2` | `3` | | | `-1` | `-2` | `-3` | |
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| = | | `1×1+2×1+3×-1` | `1×2+2×2+3×-2` | `1×3+2×3+3×-3` | | | `1×1+2×1+3×-1` | `1×2+2×2+3×-2` | `1×3+2×3+3×-3` | | | `-1×1-2×1-3×-1` | `-1×2-2×2-3×-2` | `-1×3-2×3-3×-3` | |
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| = | | `1+2-3` | `2+4-6` | `3+6-9` | | | `1+2-3` | `2+4-6` | `3+6-9` | | | `-1-2+3` | `-2-4+6` | `-3-6+9` | |
|
| = | | `0` | `0` | `0` | | | `0` | `0` | `0` | | | `0` | `0` | `0` | |
|
Here `A^2 = 0`, so `A` is a nilpotent matrix whose index is 2
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