for square matrices A and B show that [A,B]=0 then [A^2,B]=0

Solutions

Expert Solution

The symbol [,] here means the COMMUTATOR for matrices, or for two matrices A and B of suitable dimensions, [A,B]=AB-BA. The name comes from the word commute, i.e., two matrices A and B are said to commute if AB=BA. Then, if A and B commute, then [A,B]=0.

(Not to state the obvious, but all these steps are necessary since commutativity is not necessarily true in case of matrix multiplications,i.e., not all matrices A and B satisfy AB=BA even if they have the same order, are both square, etc.)

Let A and B be square matrices such that [A,B]=0, i.e., they commute. Then, .

Now, we shall multiply first from the left by A, and then we shall multiply from the right by A. These multiplications are possible since the matrices are both square matrices of the same, hence any number of multiplications on any side are possible, and product of any such multiplication will be a square matrix of the same order as the matrices A and B.

Then,

-----(1)

Also,

-----(2)

[The above two results are obtained keeping in mind the associativity of matrix multiplication.i.e, A(BC)=(AB)C].

From (1) and (2), we have

[By definition of [,] ]

Therefore, [A,B]=0 . This is the required result.

Colby Messinger answered 1 year ago

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