Final answer:
The complexity of multiplying an n x n matrix by an n x 1 vector is O(n^2), involving n multiplications and n-1 additions for each of the n rows.
Step-by-step explanation:
Complexity of Matrix-Vector Multiplication
The complexity of multiplying an n x n matrix by an n x 1 vector is governed by the number of operations required to perform the multiplication. In each row of the matrix, you perform n multiplications and n-1 additions to compute the corresponding entry in the resulting vector. Therefore, for all n rows we will perform n multiplications and n-1 additions, leading to a complexity of O(n^2).
For example, assume our matrix A represents distances (in meters) and vector x represents scalar multipliers. If each element in A is 27.5 and vector x represents a scalar multiplier of 3, the resulting multiplication would scale each distance by 3, just as one would scale the distance of a trip's leg by a scalar when multiplying a vector by a positive scalar.
Understanding these operations is critical for algorithms that involve large-scale matrix computations such as those used in computer graphics, data processing, or various types of scientific simulations. The actual complexity can vary with different matrix structures or the use of optimized algorithms like Strassen's algorithm, but the standard complexity remains O(n^2) in the general case.