Chapter 2 - Matrices and Systems of Linear Equations - 2.2 Matrix Algebra - Problems - Page 137: 36

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a) We know $A$ is an symmetric matrix, then $A+A^T$ $\rightarrow B=\frac{1}{2}(A+A^T)=\frac{1}{2}(A+A)=A$ and $C=\frac{1}{2}(A-A^T)=\frac{1}{2}(A-A)=0_n$ b) If $A$ is an $n\times n$ skew-symmetric matrix, then $A^T=-A$ we have $B=\frac{1}{2}(A+A^T)=\frac{1}{2}(A-A)=0_n$ and $C=\frac{1}{2}(A-A^T)=\frac{1}{2}(A-(-A))=A$