Chapter 2 - Matrices and Systems of Linear Equations - 2.2 Matrix Algebra - Problems - Page 136: 23

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$A=[a_{ij}]$ then $A^T=[a_{ij}]^T=[a_{ji}]$ and $(A^T)^T=[a_{ji}]^T=[a_{ij}]=A$ Let $A=[a_{ij}]$ and $C=[c_{ij}]$ $[A+C]^T=[A+C]_{ji}=[A]_{ji}+[C]_{ji}=[A^T]_{ij}+[C^T]_{ij}=[A^T+C^T]_{ij}$ $\rightarrow (A+C)^T=A^T+C^T$