Matrix Multiplication and Resultant Matrix Determination

Given matrix A which is $\begin{bmatrix}3 & 4 \\ 7 & 2 \\ 5 & 9\end{bmatrix}$ and matrix B which is $\begin{bmatrix}3 & 1 & 5 \\ 6 & 9 & 7\end{bmatrix}$ , can matrix A be multiplied by matrix B? If so, find the size and elements of the resultant matrix.

Matrix multiplication is a fundamental operation in linear algebra, which involves finding the product of two matrices. This operation is only defined when the number of columns in the first matrix matches the number of rows in the second matrix. Understanding this condition is crucial because it highlights the importance of matrix dimensions in defining valid operations. This problem involves determining whether the given matrices, A and B, can be multiplied together and, if they can, what the resultant matrix will be like in terms of its size and elements.

Once the multiplication condition is satisfied, the next step is to carry out the multiplication process. In practice, matrix multiplication involves dot products between rows of the first matrix and columns of the second matrix. For each element of the resulting matrix, we take corresponding elements from the appropriate rows and columns, multiply them pairwise, and sum them up. The resultant matrix will have a number of rows equal to the number of rows in the first matrix, and a number of columns equal to the number of columns in the second matrix.

This problem also introduces the concept of the structure of matrices and how dimensions govern not only the feasibility of operations but also the structure of potential solutions. Recognizing and computing the resultant matrix's size requires an understanding of these foundational principles, which will be beneficial for more advanced topics like solving linear systems and working with transformations in vector spaces.