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  • one year ago

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

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    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 \(A^T\)!

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