When is matrix multiplication valid?

Matrix multiplication is valid when the inner dimensions of the matrices involved match. This is a fundamental rule of matrix operations. To elaborate, if you have two matrices, let's call one matrix A with dimensions (m x n) and another matrix B with dimensions (p x q), the multiplication of A and B is possible if the number of columns in A (which is n) is equal to the number of rows in B (which is p).

When this condition is satisfied, the resulting matrix will have dimensions corresponding to the outer dimensions, specifically (m x q). Therefore, it's crucial to focus on the inner dimensions to determine if multiplication can occur. Understanding this concept is vital, as it lays the groundwork for performing further operations in linear algebra, particularly in fields such as engineering, computer science, and physics.

The other options do not apply universally to all cases of matrix multiplication. For instance, matrices do not need to be square to be multiplied, and the outer dimensions matching does not guarantee valid multiplication unless the inner dimensions match. Similarly, matrix multiplication is not always valid regardless of the dimensions involved.