pyroknife said:An n × n matrix is skew-symmetric provided
A^T = −A. Show that if A is skew-symmetric and
n is an odd positive integer, then A is not
invertible.
When you do this proof, is it necessary to prove that the determinant of A transpose = determinant of -A?
pyroknife said:An n × n matrix is skew-symmetric provided
A^T = −A. Show that if A is skew-symmetric and
n is an odd positive integer, then A is not
invertible.
When you do this proof, is it necessary to prove that the determinant of A transpose = determinant of -A?
Dick said:Why would you need to prove that? You are given that A^T=(-A), so it's pretty obvious that det(A^T)=det(-A). For a general matrix, what's the relation between det(A) and det(A^T)??
pyroknife said:Generally, it takes me a little longer to do a proof than it would for me to do a calculation (application of proof/theory). I'm not sure if this is the same with everyone. Usually I can end up figuring it out, but sometimes I'm not thinking in the right direction which can end up taking a long time.
Do you guys know of any sites that have problems where I can get some practice with matrix proofs?
Dick said:Why don't you start with this one? It's pretty easy. Answer my first question and then tell me what the relation is between det(-A) and det(A).
pyroknife said:I said it was the same in my other post.
det(-A) = (-1)^n * det(A)
using the concept where A is an nxn matrix and a is a constant. Det (-aA) = -a^n * det(A)
This is how I attempted to do this problem:
det(A^T)=det(-A)=det(A)
since n is odd det(-A) = -det(A)
-det(A)=det(A)
0 is the the only # where the negative of itself is itself.That last part is seemingly easy, but it took me a while to think in that direction.
Dick said:Ah, ok. That's what "The same" meant. You've got the proof in your head just fine. Your statement has a funny looking det(-A)=det(A) statement. I assume that's a typo. Actually I see what you mean by that. You mean det(A^T)=det(A) (because it's true for any matrix) and det(A^T)=det(-A) (because the matrix is skew). Putting all of that in one line without giving reasons is confusing and bad style.
pyroknife said:Oh yeah. I think it would've been better if I had it as det(A)=det(A^t)=det(-A) or in separate lines like you said.
Do you know of any sites where I can get more practice with proofs?
A "Matrix Proof" is a type of mathematical proof that uses matrices to demonstrate the validity of a statement or problem. It involves manipulating matrices through various operations to arrive at a solution or conclusion.
A "Matrix Proof" differs from other types of proofs because it uses matrices as the main tool for solving mathematical problems. This requires a strong understanding of matrix operations and properties.
"Matrix Proof" is commonly used in fields such as linear algebra, computer graphics, and physics. It can be applied to solve problems related to systems of linear equations, transformations, and eigenvalues.
The key steps in a "Matrix Proof" include identifying the problem or statement to be proven, setting up the necessary matrices, performing operations on the matrices, and arriving at a solution or conclusion that proves the original statement.
To improve your skills in "Matrix Proof", it is important to have a strong foundation in linear algebra and matrix operations. Practice solving problems and familiarize yourself with different matrix properties and theorems. Additionally, seeking guidance from a teacher or mentor can also help in improving your skills.