Help with determinants and eigenvalues.

Attachment 19739

For #1, I currently have that the two values are equal if A is either a matrix whose determinant is zero or a 1x1 matrix.

I don't really have any direction for the others. I see that the determinant of a square matrix whose columns add to zero is 0, but I'm not sure of why. Can it somehow be reduced to the zero matrix?

Edit: Okay, messed around with row reduction. Quickly found that all columns adding to zero forces at least one row to row-reduce to a zero row, forcing a determinant of zero.

det(cA)=c^n*det(a) where n is the number of rows so as long as you have an odd number of rows and the scalar is -1, then det(-A)=-det(A)

I'm not following.

-det(A) = det(-A) and det(-A) = -1^n*det(A) don't immediately seem to be equivalent statements. In that case, a 2x2 matrix would have the property det(A) = det(-A).

Oh, okay. Nevermind. Odd number of rows.

Quote:

---

Originally Posted by pantsaregood

I'm not following.

-det(A) = det(-A) and det(-A) = -1^n*det(A) don't immediately seem to be equivalent statements. In that case, a 2x2 matrix would have the property det(A) = det(-A).

Oh, okay. Nevermind. Odd number of rows.

---

2) If are the columns it's asking for what given that . Let me ask you this: what is the determinant of a matrix whose columns are linearly dependent equal? Moreover, how does that apply to this question?

3) This is pretty straightforward. I don't know how to give a decent hint that doesn't give it away. Merely note that and go from there.

4) If is an eigenvalue for note that ...so...