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tornado28
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I'm preparing for a qualifying exam and this problem came up on a previous qual:
Let A and B be nxn matrices. Show that if A + B = AB then AB=BA.
Let A and B be nxn matrices. Show that if A + B = AB then AB=BA.
That's only true if the two matrices are diagonalizable to begin with! :)tornado28 said:Thanks. I found another thread with information about when matrices commute. Apparently two matrices commute iff they're simultaneously diagonalizable.
A commutative matrix is a matrix that satisfies the commutative property, which states that the order of matrix multiplication does not affect the result. In other words, if matrices A and B are commutative, then A*B = B*A.
You can determine if two matrices commute by multiplying them in both orders and comparing the results. If the results are the same, then the matrices commute. Another way is to check if the matrices have the same eigenvectors and eigenvalues, as commutative matrices share the same eigendecomposition.
The commutativity of matrices is important in linear algebra as it simplifies calculations and makes certain operations, such as diagonalization, easier. It also has applications in physics, particularly in quantum mechanics.
Yes, non-square matrices can commute. However, the commutative property only applies to square matrices, so non-square matrices must have the same dimensions in order to commute.
If matrices do not commute, then the order of multiplication matters and the result will be different depending on the order. This can make calculations more complex and can also affect the interpretation of the results in certain applications.