Matrices

To multiply two matrices, take each row of the first matrix and each column of the second matrix. The entry in row $i$, column $j$ of the product is the sum of the products of corresponding entries from row $i$ of the first matrix and column $j$ of the second matrix. For example, the top-left entry of $AB$ is $a_{11}{b}_{11}+a_{12}{b}_{21}$. Matrix multiplication requires the number of columns of the first matrix to equal the number of rows of the second matrix. The transpose $A^T$ swaps rows and columns, so the entry in row $i$, column $j$ of $A^T$ is the entry in row $j$, column $i$ of $A$. The trace of a square matrix is the sum of its diagonal entries.