Chapter 2 - Matrices and Systems of Linear Equations - 2.9 Chapter Review - Additional Problems - Page 192: 12

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Let $A$ be an $n \times n$ matrix. a) Let $A$ be in a form $A=[a_{ij}]$, then $(i,j) -$ entry of the matrix $A^2$ is: $\sum^n_{k=1}a_{ik}a_{kj}$ b) Asssume $A$ is symmetric matrix, then $A=A^T$. Obtain: $(A^2)^T=(AA)^T=A^TA^T=AA=A^2$ Hence, $A^2$ is symmetric matrix.