Question

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

Answer 1

Answer:

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

$A = \left[\begin{array}{cc}1&1\\1&2\end{array}\right] \\B = \left[\begin{array}{cc}0&2\\2&0\end{array}\right]$

Then, both are symmetric, but

$AB = \left[\begin{array}{ccc}2&2\\4&2\end{array}\right]$

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 $(A+B){_i}{_j} = A_i_j + B_i_j = A_j_i + B_j_i = (A+B)_j_i$ . 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

$A = \left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] \\B = \left[\begin{array}{ccc}-1&0&0\\0&-1&0\\0&0&-1\end{array}\right]$

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.