Linear algebra-1

to find a product of a matrix $A=\begin{bmatrix} 1&2\\ 3&4 \end{bmatrix}\begin{bmatrix} 2&3\\ 5&7\end{bmatrix}$ we first want to find each individual part of the matrix, that is $A_{1,1}= \begin{bmatrix} 1&2\end{bmatrix}\begin{bmatrix} 2\\ 5 \end{bmatrix},A_{1,2}=\begin{bmatrix} 1&2\end{bmatrix}\begin{bmatrix} 3\\ 7 \end{bmatrix}, A_{2.1}=\begin{bmatrix} 3&4\end{bmatrix}\begin{bmatrix} 2\\ 5 \end{bmatrix},A_{2,2}=\begin{bmatrix} 3&4\end{bmatrix}\begin{bmatrix} 3\\ 7 \end{bmatrix}$ the form of multiplication is called a dot product and it is $\begin{bmatrix} a_1&a_2&\cdots&a_n\end{bmatrix}\begin{bmatrix} b_1\\ b_2\\.\\.\\.\\b_n \end{bmatrix}=a_1 b_1+a_2 b_2+\cdots+a_n b_n$ so we can use that to get: $A_{1,1}= 1*2+3*5=12, A_{1,2}= 1*3+2*7=17,A_{2,1}= 3*2+4*5=26,A_{2,2}= 3*3+4*7=12$ and the matrix is $A=\begin{bmatrix} 12&17\\ 26&37 \end{bmatrix}$ and the trace is $12+37=\boxed{49}$