# Show that if A is nonsingular symmetric matrix

1. Dec 2, 2003

### franz32

Hello everyone! Can anyone help me here in this theorems (prove)?
(Or solve)

1. Suppose that A and B are square matrices and AB = 0. (as in zero matrix) If B is nonsingular, find A.

2. Show that if A is nonsingular symmetric matrix, then A^-1
is symmetric.

I hope these won't bother...

2. Dec 2, 2003

Staff Emeritus
For the first one, study the definition your book or notes gives for "nonsingular". Look at it again. How does that definition bear of the product AB = 0?

For the second one, do they give you a method for finding the inverse of a square matrix? What happens in that method when aij = aji for all i and j?

3. Dec 4, 2003

### franz32

Thank you

Hello!

I got the answer... for the first, A must be a zero matrix. But
there are times when a matrix multiplied by a nonzero matrix can still result to a product of zero matrix. When is that?

For no. 2, I got it already. it's a matter of direct substitution using the properties given in my lecture.

4. Dec 4, 2003

### HallsofIvy

Well, you know that a non-zero matrix multiplied by the zero matrix gives the zero matrix so I assume you mean "a nonzero matrix multiplied by a non-zero matrix" can give a zero matrix.

You have just determined that if one of the matrices is non-singular, then the other must be the zero matrix. What happens if you multipy two singular matrices?

I think you will find that true for many problems! Amazing that a lecture (and text book) can actually be useful isn't it?

5. Dec 5, 2003

### franz32

I got it

Hello.

You mean that the product of 2 singular (square ) matrices must result to either another matrix or a zero matrix?

Oh, thank you.

6. Dec 6, 2003

### HallsofIvy

"either another matrix or the zero matrix"???

Well that's always true isn't it!

What I meant was that, since you have already proved that if AB= O with A non-singular then B=0, the ONLY way you could get AB= 0 without either A or B 0 is to multiply two singular matrices.
The product of two singular matrices is not always 0 but is always a singular matrix.

7. Dec 12, 2003

### franz32

Hello

Hi again.

Thank you very much! =)