Question

Express the matrix \(B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\) in the form of the sum of a symmetric matrix and a skew-symmetric matrix.

Hint:

Use P=(B+B^T)/2 for the symmetric part and Q=(B-B^T)/2 for the skew-symmetric part.

Answer 1

Step 1: Verifying P is symmetric:
Check \(P^T=P\) entrywise: the off-diagonal pairs \((1,2)=(2,1)=-3/2\), \((1,3)=(3,1)=-3/2\), \((2,3)=(3,2)=1\) all match, confirming symmetry.

Step 2: Verifying Q is skew-symmetric:
Check \(Q^T=-Q\): diagonal entries are all \(0\), and off-diagonal pairs are negatives of each other, e.g. \(Q_{12}=-1/2=-Q_{21}=-(1/2)\), confirming skew-symmetry.

Step 3: Confirming P+Q reproduces B:
Adding entrywise, e.g. position (1,1): \(2+0=2\)✓; position(1,2): \(-3/2-1/2=-2\)✓; position(2,1): \(-3/2+1/2=-1\)✓ — matches every entry of the original \(B\).

Final Answer:
\[ \boxed{B=P+Q} \]