1. is Idempotent Matrix ?
`[[2,-2,-4],[-1,3,4],[1,-2,-3]]`
Solution:
A square matrix `A` is called an idempotent matrix, if `A^2 = A`.
| `A` | = | | `2` | `-2` | `-4` | | | `-1` | `3` | `4` | | | `1` | `-2` | `-3` | |
|
| `A×A` | = | | `2` | `-2` | `-4` | | | `-1` | `3` | `4` | | | `1` | `-2` | `-3` | |
| × | | `2` | `-2` | `-4` | | | `-1` | `3` | `4` | | | `1` | `-2` | `-3` | |
|
| = | | `2×2-2×-1-4×1` | `2×-2-2×3-4×-2` | `2×-4-2×4-4×-3` | | | `-1×2+3×-1+4×1` | `-1×-2+3×3+4×-2` | `-1×-4+3×4+4×-3` | | | `1×2-2×-1-3×1` | `1×-2-2×3-3×-2` | `1×-4-2×4-3×-3` | |
|
| = | | `4+2-4` | `-4-6+8` | `-8-8+12` | | | `-2-3+4` | `2+9-8` | `4+12-12` | | | `2+2-3` | `-2-6+6` | `-4-8+9` | |
|
| = | | `2` | `-2` | `-4` | | | `-1` | `3` | `4` | | | `1` | `-2` | `-3` | |
|
Here `A^2 = A`, so `A` is an idempotent matrix
2. is Idempotent Matrix ?
`[[1,-2,-4],[-1,3,4],[1,-2,-3]]`
Solution:
A square matrix `A` is called an idempotent matrix, if `A^2 = A`.
| `A` | = | | `1` | `-2` | `-4` | | | `-1` | `3` | `4` | | | `1` | `-2` | `-3` | |
|
| `A×A` | = | | `1` | `-2` | `-4` | | | `-1` | `3` | `4` | | | `1` | `-2` | `-3` | |
| × | | `1` | `-2` | `-4` | | | `-1` | `3` | `4` | | | `1` | `-2` | `-3` | |
|
| = | | `1×1-2×-1-4×1` | `1×-2-2×3-4×-2` | `1×-4-2×4-4×-3` | | | `-1×1+3×-1+4×1` | `-1×-2+3×3+4×-2` | `-1×-4+3×4+4×-3` | | | `1×1-2×-1-3×1` | `1×-2-2×3-3×-2` | `1×-4-2×4-3×-3` | |
|
| = | | `1+2-4` | `-2-6+8` | `-4-8+12` | | | `-1-3+4` | `2+9-8` | `4+12-12` | | | `1+2-3` | `-2-6+6` | `-4-8+9` | |
|
| = | | `-1` | `0` | `0` | | | `0` | `3` | `4` | | | `0` | `-2` | `-3` | |
|
Here `A^2 != A`, so `A` is not an idempotent matrix
This material is intended as a summary. Use your textbook for detail explanation.
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