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Find A = [[1,2],[3,4]], B = [[2,-1],[-1,2]], (A * B)' = B' * A'

Solution:
Your problem `->` A = [[1,2],[3,4]], B = [[2,-1],[-1,2]], (A * B)' = B' * A'


To find LHS = `(A × B)^T`

`A×B`=
`1``2`
`3``4`
×
`2``-1`
`-1``2`


=
`1×2+2×-1``1×-1+2×2`
`3×2+4×-1``3×-1+4×2`


=
`2-2``-1+4`
`6-4``-3+8`


=
`0``3`
`2``5`


`(A × B)^T` = 
`0``3`
`2``5`
T
 = 
`0``2`
`3``5`


`:.``(A × B)^T` = 
`0``2`
`3``5`
` ->(1)`


To find RHS = `(B^T) × (A^T)`

`B^T` = 
`2``-1`
`-1``2`
T
 = 
`2``-1`
`-1``2`


`A^T` = 
`1``2`
`3``4`
T
 = 
`1``3`
`2``4`


`(B^T)×(A^T)`=
`2``-1`
`-1``2`
×
`1``3`
`2``4`


=
`2×1-1×2``2×3-1×4`
`-1×1+2×2``-1×3+2×4`


=
`2-2``6-4`
`-1+4``-3+8`


=
`0``2`
`3``5`


`:.``(B^T) × (A^T)` = 
`0``2`
`3``5`
` ->(2)`


From (1) and (2)
`:. (A × B)^T=(B^T) × (A^T)` (proved)






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