Matrix Problems

[kennyken]

The following is a list of statements regarding n x n nonsingular matrices. In each case, either prove that the statement is generally true or find 2 x 2 matrices for which it is false.

i)If A and B are non-singular, then so is A + B
ii)If A and B are non-singular and A^2B^2=I then (AB)^-1=BA
iii)The only n x n non-singular reduced row echelon matrix is I.
iv)If S is non-singular and symmetric (that is, S=S(Transposed)), then S^-1 is also symmetric
v)If K is non-singular and skew-symmetric, then K^-1 is skew symmetric.
vi)I is the only n x n non-singular idempotent matrix
vii)An n x n nilpotent matrix must be singular.

I know this is difficult if you don't remember much about matrices, so if you have any questions concerning the problem you can ask and I will try to clarify. But I really really need help, this is really bugging me.

non-singular means that the matrix has an inverse.

Thanks

 

[Gene]

i) I offer
[1,2],[3,4] inverse [-2,1],[1.5,-5] and
[4,3],[2,1] inverse [-.5,1.5],[1,-2]
[1,2],[3,4] + [4,3],[2,1] = [5,5],[5,5] no inverse
But I am confused 'cause these both reduce to identity, the same as when I picked on Soroban's answers. Does the clarification you offered cover whether a reduced matrix is the same as it was before reduction? If I do reduce them they are their own inverses and the sum does have an inverse. I need to understand this before I can think about the others.

 

[pka]

Note that: “list of statements regarding n x n nonsingular matrices list of statements regarding n x n nonsingular matrices.”

Gene also, I+(-I)=0, which is singular. So (i) is false.

Depending upon the strictness of the definition “reduced row echelon matrix”
I think that (iii) is true by default.

Now we know that (A<SUP>-1</SUP>)<SUP>n</SUP>=(A<SUP>n</SUP>)<SUP>-1</SUP>.
What does that say about non-singular nilpotent matrix?