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 Solution provided by AtoZmath.com
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        | Prove that any two matrix expression is equal or not calculator |  
        |  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` |  | 
 | 
 | = | |  | `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` |  | 
 | 
 | = | |  | `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` |  | 
 | 
 | `:.` | `(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` |  | 
 | 
 | `A^T` | = | | |  | `3` | `1` | `1` |  |  |  | `-1` | `2` | `1` |  |  |  | `1` | `1` | `1` |  | 
 | T | 
 | = | |  | `3` | `-1` | `1` |  |  |  | `1` | `2` | `1` |  |  |  | `1` | `1` | `1` |  | 
 | 
 | `(B^T)×(A^T)` | = | |  | `5` | `7` | `1` |  |  |  | `0` | `-6` | `1` |  |  |  | `-2` | `0` | `2` |  | 
 | × | |  | `3` | `-1` | `1` |  |  |  | `1` | `2` | `1` |  |  |  | `1` | `1` | `1` |  | 
 | 
 | = | |  | `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` |  | 
 | 
 | = | |  | `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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