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Chapter 2: Problem 62

Prove that if \(A\) and \(B\) are \(n \times n\) skew-symmetric matrices, then \(A+B\) is skew-symmetric.

Short Answer

Expert verified
The sum of two skew-symmetric matrices \(A\) and \(B\) is also skew-symmetric because \((A+B)^T = -(A+B)\). This is the result of substituting the skew-symmetry property of \(A\) and \(B\) into the expression for the transpose of \(A+B\).

Step by step solution

01

Write down the formula for a skew-symmetric matrix

A matrix \(A\) is skew-symmetric if and only if \(A^T = -A\). Similarly, matrix \(B\) is skew-symmetric if \(B^T = -B\). This is what we start with.
02

Write down what we want to prove

We want to prove that \(A + B\) is also skew-symmetric. This would be true if and only if \((A+B)^T = -(A + B)\).
03

Use the properties of transpose

Using the properties of transpose, we know that the transpose of the sum of \(A\) and \(B\) is equal to the sum of the transpose of \(A\) and the transpose of \(B\). So, \((A+B)^T = A^T + B^T\).
04

Substitute the skew-symmetry property of A and B into the above expression

Substituting \(A^T = -A\) and \(B^T = -B\), we get \((A+B)^T = -A - B = -(A+B)\). So, \((A+B)^T = -(A+B)\)
05

Conclude the proof

Since \((A+B)^T = -(A+B)\), we can conclude that \(A + B\) is a skew-symmetric matrix if both \(A\) and \(B\) are skew-symmetric matrices.

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