Given two matrices:
$$ A= \begin{pmatrix} 1 & 0 & 1 & 2\\ 2 & 1 & -1 & 1\\ 0 & 1 & 0 & 2\\ \end{pmatrix} $$
$$ B= \begin{pmatrix} 2 & -1 & -2 \\ 3 & 4 & -3 \\ 0 & 1 & 2 \\ \end{pmatrix} $$
a) Construct two symmetric square matrices from matrix $A$
b) Calculate the symmetric and anti-symmetric part of matrix $B$
I don't know how to go about question a). I tried a LU Decomposition but it didn't give the desired outcome. Only other thing I could think of was to take the Lower matrix from the LU decomposition, then add it to its transpose, which of course gives a symmetric square matrix but it seemed too "forced" to me, plus it only gives one matrix instead of the two that are demanded. So I'm sure that there's another method that I'm not aware of.
As for question b) I don't understand what they mean by "symmetric and anti-symmetric parts" and what does it mean to calculate them.
Hope someone can help me solve this, or at least give me some hints. Thanks.
For b): Prove that $B= \frac{1}{2}(B+B^t)+\frac{1}{2}(B-B^t)$
This decomposition can be applied to all square matrices. What are the properties of the matrices that are summated?