Determinant Properties

[hkus10]

1) If det(AB) = 0, is det(A) or det(B) = 0? Give reasons for your answer.

Q1) First, cannot both det(A) or det(B) be 0? If it can, is this statement false. In any case, how can I prove that this is true for all statement since I only know how to find an example to show this is true, which cannot represent all the possibility.

2) Show that if A is singular and Ax = b, b is not equal to 0, has one solution, then it has infinitely many.

Q2) How to approach this question?

3) Let A^2 = A. Prove that either A is singular or det(A) = 1.

Q3) How can I approach this question?

 

[Robert1986]

1) If det(AB) = 0, is det(A) or det(B) = 0? Give reasons for your answer.

Q1) First, cannot both det(A) or det(B) be 0? If it can, is this statement false. In any case, how can I prove that this is true for all statement since I only know how to find an example to show this is true, which cannot represent all the possibility.

First, if det(A) = det(B) = 0, then the statement is most certainly true. "Or" does not mean "one or the other, but not both", it means at least one. What do you know about dets? For example, can you use the fact the det(AB)=det(A)det(B), if you can, this should be easy, so you probably can't. But, tell us what you are using as your definition of det, because there are various ways to define it.

[QUOTE
2) Show that if A is singular and Ax = b, b is not equal to 0, has one solution, then it has infinitely many.

Q2) How to approach this question?[/QUOTE]
How does singularity relate to the det? Work with that to see if it leads anywhere.

3) Let A^2 = A. Prove that either A is singular or det(A) = 1.

Q3) How can I approach this question?

Again, how does singularity (what does that mean, first of all) relate to determinents? That is if A is singular, what can you tell about its det?