Prove or Disprove: If AB=-BA, A or B is Singular

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Start date Oct 6, 2011
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Matrices

In summary, the statement "AB=-BA" means that the product of matrix A and matrix B is equal to the negative product of matrix B and matrix A. A matrix is singular if its determinant is equal to 0. This means that the matrix does not have an inverse and cannot be inverted. However, the statement "AB=-BA" does not always imply that either A or B is singular. It is possible for both A and B to be non-singular matrices and still satisfy this equation. For example, A = [1 2; 3 4] and B = [5 6; 7 8]. Additionally, if one of the matrices, say A, is a zero matrix, then A will

Oct 6, 2011

fofo

Given any two n X n matrices A and B , prove or disprove the following statements:
If AB=-BA then at least one of A and B is singular.

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Oct 6, 2011

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Related to Prove or Disprove: If AB=-BA, A or B is Singular

1. What does the statement "AB=-BA" mean?

The statement "AB=-BA" means that the product of matrix A and matrix B is equal to the negative product of matrix B and matrix A.

2. What does it mean for a matrix to be singular?

A matrix is singular if its determinant is equal to 0. This means that the matrix does not have an inverse and cannot be inverted.

3. Does the statement "AB=-BA" always imply that either A or B is singular?

No, the statement "AB=-BA" does not always imply that either A or B is singular. It is possible for both A and B to be non-singular matrices and still satisfy this equation.

4. Can you provide an example where A and B are both non-singular matrices but still satisfy the equation "AB=-BA"?

Yes, consider the matrices A = [1 2; 3 4] and B = [5 6; 7 8]. Both A and B are non-singular matrices and their product AB = [19 22; 43 50] is equal to the negative product of BA = [-23 -34; -31 -46]. Therefore, AB=-BA is satisfied.

5. Is there a scenario where either A or B is singular and the equation "AB=-BA" is still true?

Yes, if one of the matrices, say A, is a zero matrix, then A will be singular. In this case, the equation "AB=-BA" will hold true regardless of the singularity of B.