Express the matrix A = [(4, 2, -1), (3, 5, 7), (1, -2, 1)] as the sum of a symmetric and skew-symmetric matrix.

Question

Express the matrix A = \(\begin{bmatrix} 4 & 2 & -1 \\ 3 & 5 & 7 \\ 1 & -2 & 1\end{bmatrix}\)as the sum of a symmetric and skew-symmetric matrix.

Answer 1

Any square matrix A can be expressed as the sum of a symmetric and skew-symmetric, i.e., A = \(\frac{A + A'}{2} + \frac{A - A'}{2}\), where \(\frac{A + A'}{2} \) and \(\frac{A - A'}{2}\) are symmetric and skew-symmetric matrices respectively.

Thus, matrix A is expressed as the sum of symmetric matrix and skew-symmetric matrix.