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Home > Matrix & Vector calculators > Prove that any two matrix expression is equal or not example
9. Prove that any two matrix expression is equal or not example ( Enter your problem )
( Enter your problem )
  1. Example-1
  2. Examples-2
 		Other related methods
  1. Addition of two matrix
  2. Multiplication of two matrix
  3. Division of two matrix
  4. Power of a matrix
  5. Transpose of a matrix
  6. Determinant of a matrix
  7. Adjoint of a matrix
  8. Inverse of a matrix
  9. Prove that any two matrix expression is equal or not
  10. Minor of a matrix
  11. Cofactor of a matrix
  12. Trace of a matrix
8. Inverse of a matrix
(Previous method)		2. Examples-2
(Next example)

1. Example-1


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`


=
`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`


`(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`


=
`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`


`:.``(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)

This material is intended as a summary. Use your textbook for detail explanation.
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8. Inverse of a matrix
(Previous method)		2. Examples-2
(Next example)


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