These are True/False concept questions from Howard Anton's Linear Algebra, 11th edition, contained in the first 200 pages of the book. I did some 60 of them and I got these wrong and I'm not sure what I'm doing wrong:
I will continually delete questions as I get answers
5. If A and B are $nXn$ matrices such that $A + B$ is symmetric, then A and B are symmetric.
6. If A is a 4x4 matrix and B is obtained from A by swap of the first two lines, followed by the swap of the last two lines, then $det(B) = det(A)$. The official answer is False.
7. If A is a 3x3 matrix and B is a matrix obtained from A by multiplying the first line by 4, then the last by 3/4, then $det(B) = 3det(A)$. The official answer is False.
8. If $A²$ is a symmetric matrix, then A is symmetric.
5.I don't know how I would prove that. If I start creating two matrices A and B, both symmetric, then I can show that A + B is symmetric, but the problem is showing the other way around, that is, you need to come up with one symmetric matrix and decompose it into two symmetric matrices and I don't know how to do that.
6.Yes, swap two lines and the sign of the determinant changes. Swap two lines again and the sign of the determinant comes back just as in the original matrix.
7.Yes, in calculating the determinant you could factor out a 4, then factor out a 3/4. Then $4*(3/4) = 3$.
8.I can start with a symmetric matrix and show that if you square it, A² is symmetric, but then again, what I need is the other way around, and I don't know how to proceed here.