OpenStudy (anonymous):

A matrix A is skew symmetric if A^T=-A. If A is a skew symmetric matrix, what type of matrix is A^T?

2 years ago
OpenStudy (empty):

Well, let's go ahead and rename our matrix $$A^T$$ so that we can think about it a little easier maybe? Let's call it B. $A^T=B$ Now to see if B is skew symmetric or not, let's plug this into the equation that all skew symmetric matrices must satisfy. If we end up with a true statement then we have shown that it is indeed skew symmetric: $B^T=-B$ Now we can plug in B again now that we're not distracted, and follow through with the algebra: $(A^T)^T = -A^T$ Now we know that the transpose of the transpose of a matrix just undoes it, so the left side's transposes cancel leaving us with just A there. $A = -A^T$ What about the right side? Well remember they said $$A^T=-A$$ so we can replace it: $A=-(-A)$ So we see that we get a true statement, therefore if $$A$$ is skew symmetric so is $$A^T$$!

2 years ago