1. is Involutary Matrix ?
`[[-5,-8,0],[3,5,0],[1,2,-1]]`Solution:A square matrix `A` is called an involutary matrix, if `A^2 = I` where `I` is the identity matrix.
| `A` | = | | `-5` | `-8` | `0` | | | `3` | `5` | `0` | | | `1` | `2` | `-1` | |
|
| `A×A` | = | | `-5` | `-8` | `0` | | | `3` | `5` | `0` | | | `1` | `2` | `-1` | |
| × | | `-5` | `-8` | `0` | | | `3` | `5` | `0` | | | `1` | `2` | `-1` | |
|
| = | | `-5×-5-8×3+0×1` | `-5×-8-8×5+0×2` | `-5×0-8×0+0×-1` | | | `3×-5+5×3+0×1` | `3×-8+5×5+0×2` | `3×0+5×0+0×-1` | | | `1×-5+2×3-1×1` | `1×-8+2×5-1×2` | `1×0+2×0-1×-1` | |
|
| = | | `25-24+0` | `40-40+0` | `0+0+0` | | | `-15+15+0` | `-24+25+0` | `0+0+0` | | | `-5+6-1` | `-8+10-2` | `0+0+1` | |
|
| = | | `1` | `0` | `0` | | | `0` | `1` | `0` | | | `0` | `0` | `1` | |
|
Here `A^2 = I`, so `A` is an involutary matrix
2. is Involutary Matrix ?
`[[2,-8,0],[3,5,0],[1,2,-1]]`Solution:A square matrix `A` is called an involutary matrix, if `A^2 = I` where `I` is the identity matrix.
| `A` | = | | `2` | `-8` | `0` | | | `3` | `5` | `0` | | | `1` | `2` | `-1` | |
|
| `A×A` | = | | `2` | `-8` | `0` | | | `3` | `5` | `0` | | | `1` | `2` | `-1` | |
| × | | `2` | `-8` | `0` | | | `3` | `5` | `0` | | | `1` | `2` | `-1` | |
|
| = | | `2×2-8×3+0×1` | `2×-8-8×5+0×2` | `2×0-8×0+0×-1` | | | `3×2+5×3+0×1` | `3×-8+5×5+0×2` | `3×0+5×0+0×-1` | | | `1×2+2×3-1×1` | `1×-8+2×5-1×2` | `1×0+2×0-1×-1` | |
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| = | | `4-24+0` | `-16-40+0` | `0+0+0` | | | `6+15+0` | `-24+25+0` | `0+0+0` | | | `2+6-1` | `-8+10-2` | `0+0+1` | |
|
| = | | `-20` | `-56` | `0` | | | `21` | `1` | `0` | | | `7` | `0` | `1` | |
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`A^2 != I`, so `A` is not an involutary matrix
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