D) if A and B are symmetric, is AB symmetric? Prove or give an example where it fails to be true. e) If A and B are symmetric is A+B symmetric? Prove or give an example where it fails to be true. f) Say whether the following product AB is singular, non-singular, or depends on the matrices. Explain why. i) A and B are non-singular i) A and B are singular ii) A is singular and B is non-singular g) If A and B are non-singular, is A+B non singular? Prove or give an example where it fails to be true

d) Not necessarily

e) True

f)

i)  AB is non singular

ii) AB is singular

iii)  AB is singular

g) Not necessarily

Step-by-step explanation:

d) Not neccesarily, if we take

Then, both are symmetric, but

Is not, because (AB)₁₂ = 2 and (AB)₂₁ = 4.

e) This is True. In the ith row and the jth column, since A and B are symmetric, we have . Therefore, A+B is symmetric.

f) Recall that a matrix C is non singular if and only if Det(C) = 0. Also, det(AB) = Det(A)*Det(B), therefore, if one of them is singular, then there would be a 0 in the product, and as a consequence, AB would have determinant 0, from which we can conclude that AB is singular. On the other hand, if Det(A) and Det(B) are both different from 0, then Det(AB) ≠ 0, so we can conclude that, if both A and B are non singular, then AB also is not singular. As a consequence,

i)  AB is non singular

ii) AB is singular

iii)  AB is singular

g) False, if

Then neither A nor B are singular, however the sum is the null matrix, therefore it is singular in spite of being a sum of 2 non singular matrix.