Is Every Skew Symmetric Matrix Invertible When Added to the Identity Matrix?

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In summary, we have discussed how to prove that a skew symmetric nxn matrix A with entries in R satisfies AT=-A, and have also explored how to prove that the matrix (I+A) is invertible. By using the fact that uTAu=0 for every u in R, we can show that the kernel of I+A is trivial, meaning that I+A is an injective operator and thus invertible.
  • #1
weasel3000
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A is a skew symmetric nxn matrix with entries in R. AT=-A.
1). Prove uTAu=0 for every u in R. (I made it)
2). Prove In + A is a invertible matrix.
I have no idea how to prove the 2nd.
Thanks
 
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  • #2
Hmm, I guess one of your classmates also asked this question recently, cause I can remember it.

Anyway, I'll give a hint. For which vectors u, does (I+A)u=0. Use (1) for this.
 
  • #3
micromass said:
Hmm, I guess one of your classmates also asked this question recently, cause I can remember it.

Anyway, I'll give a hint. For which vectors u, does (I+A)u=0. Use (1) for this.

? not get it. 0*u=0; (I+A)*0=0; non-zero vector *non-zero vector=0 also possible.
 
  • #4
You're right, if u=0, then (I+A)u=0. Are there other possibilities for u?
 
  • #5
thank you for your hint. However I am so dumb ,and unable to get it. R u going to get det(I+A) not 0 or find (I+A)^-1
 
  • #6
No, we are trying to establish that the kernel of I+A is trivial. This implies that I+A is an injective operator. And by some theorem of linear algebra, this implies that I+A is invertible...
 
  • #7
micromass said:
No, we are trying to establish that the kernel of I+A is trivial. This implies that I+A is an injective operator. And by some theorem of linear algebra, this implies that I+A is invertible...

oh my god. i am going to find a place to cry. i just know the null space. never learn what kernel and injective operators are. Thank you again. I am going to learn that damn THINGS tonight.
 
  • #8
Don't worry :smile: The kernel is exactly thesame as the null space. And injective (or one-to-one or monomorphism) just means that the null space is trivial.
 

FAQ: Is Every Skew Symmetric Matrix Invertible When Added to the Identity Matrix?

1. What is an invertible matrix?

An invertible matrix is a square matrix that has a unique inverse matrix. This means that when the original matrix is multiplied by its inverse, the result is the identity matrix.

2. How can I prove that a matrix is invertible?

A matrix can be proven to be invertible by showing that its determinant is non-zero. A non-zero determinant indicates that the matrix has a unique solution, and therefore has an inverse.

3. Can all matrices be inverted?

No, not all matrices can be inverted. Only square matrices (same number of rows and columns) can be inverted, and only if their determinant is non-zero.

4. Why is it important to prove that a matrix is invertible?

Invertible matrices have many applications in mathematics and science, such as solving systems of equations, finding the inverse of a transformation, and calculating the eigenvalues and eigenvectors of a matrix. Proving that a matrix is invertible ensures that these calculations and applications are accurate.

5. How can I use the inverse of a matrix in practical applications?

The inverse of a matrix can be used to solve systems of equations, find the inverse of a transformation, and calculate eigenvalues and eigenvectors. It is also useful in computer graphics, robotics, and cryptography.

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