For real symmetric matrices A and B, which of the following is true?a)AB is symmetric matrixb)AB = BAc)All eigen values of AB are real if AB = BAd)AB is invertible if either A is invertible or B is invertible.Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Introduction:
In this question, we are given that A and B are real symmetric matrices. We need to determine which of the given options is true.

Explanation:
Let's go through each option one by one and determine whether it is true or false.

Option a) AB is a symmetric matrix:
To determine whether AB is a symmetric matrix, we need to check if (AB)^T = AB, where (AB)^T represents the transpose of AB.

Taking the transpose of AB, we have (AB)^T = B^T * A^T

Since A and B are symmetric matrices, we have A^T = A and B^T = B.

Therefore, (AB)^T = B^T * A^T = BA

So, if AB = BA, then AB is a symmetric matrix. However, in general, AB may not be equal to BA. Hence, option a) is false.

Option b) AB = BA:
To determine whether AB = BA, we need to check if AB and BA are equal matrices.

Since we do not have any information about the commutativity of A and B, we cannot conclude that AB = BA. Hence, option b) is not necessarily true.

Option c) All eigenvalues of AB are real if AB = BA:
To determine whether all eigenvalues of AB are real, we need to check if AB is a Hermitian matrix.

A Hermitian matrix is a square matrix that is equal to its conjugate transpose. In the case of real matrices, the conjugate transpose is the same as the transpose.

If AB = BA, then AB is a Hermitian matrix and all its eigenvalues are real. Hence, option c) is true.

Option d) AB is invertible if either A is invertible or B is invertible:
To determine whether AB is invertible, we need to check if AB has an inverse.

In general, the product of two invertible matrices is also invertible. Therefore, if A and B are invertible matrices, then AB is also invertible.

However, if either A or B is not invertible, then AB may or may not be invertible. Hence, option d) is not necessarily true.

Conclusion:
From the above analysis, we can conclude that option c) is true. All eigenvalues of AB are real if AB = BA.

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let A = [ aij]be a 3x3 invertible matrix with real entries and B = [bij] be a matrix which is formed such that bij is the sum of all the elements except aij in the ith row. Answer the following:If there exist a matrix X with Constant elements such that A X = B, then X is

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