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Prove that any two matrix expression is equal or not calculator
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1. `[[2,3,1],[0,5,6],[1,1,2]]`
2. `[[2,1,-1],[1,0,-1],[1,1,2]]`
3. `[[3,1,1],[-1,2,1],[1,1,1]]`
4. `[[5,6,1],[0,2,3],[1,1,2]]`
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Example1. Find `(A × B)' = B' × A'` ... `A=[[3,1,1],[-1,2,1],[1,1,1]]`,`B=[[5,0,-2],[7,-6,0],[1,1,2]]`Solution:To find LHS = `(A × B)^T` `A×B` | = | | `3` | `1` | `1` | | | `-1` | `2` | `1` | | | `1` | `1` | `1` | |
| × | | `5` | `0` | `-2` | | | `7` | `-6` | `0` | | | `1` | `1` | `2` | |
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= | | `3×5+1×7+1×1` | `3×0+1×-6+1×1` | `3×-2+1×0+1×2` | | | `-1×5+2×7+1×1` | `-1×0+2×-6+1×1` | `-1×-2+2×0+1×2` | | | `1×5+1×7+1×1` | `1×0+1×-6+1×1` | `1×-2+1×0+1×2` | |
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= | | `15+7+1` | `0-6+1` | `-6+0+2` | | | `-5+14+1` | `0-12+1` | `2+0+2` | | | `5+7+1` | `0-6+1` | `-2+0+2` | |
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= | | `23` | `-5` | `-4` | | | `10` | `-11` | `4` | | | `13` | `-5` | `0` | |
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`(A × B)^T` | = | | `23` | `-5` | `-4` | | | `10` | `-11` | `4` | | | `13` | `-5` | `0` | |
| T |
| = | | `23` | `10` | `13` | | | `-5` | `-11` | `-5` | | | `-4` | `4` | `0` | |
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`:.` | `(A × B)^T` | = | | `23` | `10` | `13` | | | `-5` | `-11` | `-5` | | | `-4` | `4` | `0` | |
| ` ->(1)` |
To find RHS = `(B^T) × (A^T)` `B^T` | = | | `5` | `0` | `-2` | | | `7` | `-6` | `0` | | | `1` | `1` | `2` | |
| T |
| = | | `5` | `7` | `1` | | | `0` | `-6` | `1` | | | `-2` | `0` | `2` | |
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`A^T` | = | | `3` | `1` | `1` | | | `-1` | `2` | `1` | | | `1` | `1` | `1` | |
| T |
| = | | `3` | `-1` | `1` | | | `1` | `2` | `1` | | | `1` | `1` | `1` | |
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`(B^T)×(A^T)` | = | | `5` | `7` | `1` | | | `0` | `-6` | `1` | | | `-2` | `0` | `2` | |
| × | | `3` | `-1` | `1` | | | `1` | `2` | `1` | | | `1` | `1` | `1` | |
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= | | `5×3+7×1+1×1` | `5×-1+7×2+1×1` | `5×1+7×1+1×1` | | | `0×3-6×1+1×1` | `0×-1-6×2+1×1` | `0×1-6×1+1×1` | | | `-2×3+0×1+2×1` | `-2×-1+0×2+2×1` | `-2×1+0×1+2×1` | |
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= | | `15+7+1` | `-5+14+1` | `5+7+1` | | | `0-6+1` | `0-12+1` | `0-6+1` | | | `-6+0+2` | `2+0+2` | `-2+0+2` | |
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= | | `23` | `10` | `13` | | | `-5` | `-11` | `-5` | | | `-4` | `4` | `0` | |
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`:.` | `(B^T) × (A^T)` | = | | `23` | `10` | `13` | | | `-5` | `-11` | `-5` | | | `-4` | `4` | `0` | |
| ` ->(2)` |
From (1) and (2) `:. (A × B)^T=(B^T) × (A^T)` (proved)
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