For any positive integer n and matrix A and matrix B
Is it (AB)^n=(A^n)(B^n)?
I know that (AB)^t=(B^t)(A^t) [where t means transpose]
Is it also true for the index one ?
Fundamental Matrix Question
2009-01-27 8:36 pm
回答 (3)
2009-01-28 12:26 am
✔ 最佳答案
It is not true that (AB)n = (An)(Bn).Just take a couter-example as follows:
圖片參考：http://i182.photobucket.com/albums/x4/A_Hepburn_1990/A_Hepburn01Jan271625.jpg?t=1233044727
Obviously they are not the same. Hence, this statement is not true.
2009-01-27 16:26:20 補充：
"counter-example"
(Typing mistake)
參考: Myself~~~
2009-01-28 7:54 pm
If two matrice satisfy the commutative law of multiplication: AB=BA,
then satisfy all multiplicative identities, e.g.
A^2 - B^2 = (A+B)(A-B)
(A+B)^2 = A^2 + 2AB + B^2, etc.
then satisfy all multiplicative identities, e.g.
A^2 - B^2 = (A+B)(A-B)
(A+B)^2 = A^2 + 2AB + B^2, etc.
2009-01-28 12:55 am
Audrey Hepburn's counter-example is very clear in showing that the proposition is not true~
in fact, the proposition holds only for a certain kind of matrix, such as diagonal matrix.
2009-01-27 16:56:05 補充：
As for diagonal matrix, the multiplication is commutative: (A)(B) = (B)(A).
Note that the proposition is also true for matrix-inverseMatrix pairs.
in fact, the proposition holds only for a certain kind of matrix, such as diagonal matrix.
2009-01-27 16:56:05 補充：
As for diagonal matrix, the multiplication is commutative: (A)(B) = (B)(A).
Note that the proposition is also true for matrix-inverseMatrix pairs.
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