Show that if A is nonsingular symmetric matrix

In summary, the first theorem says that if A is nonsingular and symmetric then A^-1 is symmetric. The second theorem states that if A is a square matrix and B is nonsingular, then B^-1 is symmetric.
  • #1
franz32
133
0
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...
 
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  • #2
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
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
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?
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?

For no. 2, I got it already. it's a matter of direct substitution using the properties given in my lecture.
I think you will find that true for many problems! Amazing that a lecture (and textbook) can actually be useful isn't it?
 
  • #5
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
"either another matrix or the zero matrix"?


Well that's always true isn't it! :smile:

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
Hello

Hi again.

Thank you very much! =)
 

1. What does it mean for a matrix to be nonsingular?

A matrix is considered nonsingular if it has an inverse, meaning it can be multiplied by another matrix to produce the identity matrix, which has 1s on the diagonal and 0s everywhere else.

2. How do you determine if a matrix is symmetric?

A matrix is symmetric if it is equal to its own transpose. This means that the elements above the main diagonal are equal to the elements below the main diagonal.

3. Why is it important to know if a matrix is nonsingular and symmetric?

Knowing if a matrix is nonsingular and symmetric is important because it allows us to use certain properties and methods that are specific to these types of matrices. For example, nonsingular matrices are invertible, which is useful for solving systems of equations, and symmetric matrices have real eigenvalues and orthogonal eigenvectors, which are important for diagonalization and other applications.

4. What is the significance of proving that A is nonsingular symmetric?

Proving that A is nonsingular symmetric allows us to confidently use the properties and methods mentioned above. It also allows us to make conclusions about the matrix and its relationship to other matrices, and to use it in various applications such as optimization, statistics, and physics.

5. How do you show that A is nonsingular symmetric?

To show that A is nonsingular symmetric, we need to prove two things: 1) A is invertible, meaning it has an inverse matrix, and 2) A is equal to its own transpose. This can be done through various methods such as Gaussian elimination, diagonalization, or using the properties of determinants and eigenvectors.

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