What is the time and space complexity of the naive matrix multiplication algorithm for two n x n matrices?

What is the time and space complexity of the naive matrix multiplication algorithm for two n x n matrices?

• A. Time O(n^3); Space O(n^2)
• B. Time O(n^2); Space O(n)
• C. Time O(n); Space O(1)
• D. Time O(n^4); Space O(n^2)

Answer: (A)

The main idea is that naive matrix multiplication computes a dot product for every entry of the result matrix. For two n by n matrices, there are n^2 entries in the output, and to get each one you sum n products (each is a multiplication of a row element with a column element). That means you perform roughly n × n × n = n^3 multiplications and a similar number of additions. So the time grows cubically with n: O(n^3). For space, you need to store the resulting n by n matrix, which has n^2 entries. The input matrices already occupy space, but the amount of memory that scales with n due to the computation (the auxiliary/needed space aside from the inputs) is O(n^2). This is why the common answer is time O(n^3) and space O(n^2).

The main idea is that naive matrix multiplication computes a dot product for every entry of the result matrix. For two n by n matrices, there are n^2 entries in the output, and to get each one you sum n products (each is a multiplication of a row element with a column element). That means you perform roughly n × n × n = n^3 multiplications and a similar number of additions. So the time grows cubically with n: O(n^3).

For space, you need to store the resulting n by n matrix, which has n^2 entries. The input matrices already occupy space, but the amount of memory that scales with n due to the computation (the auxiliary/needed space aside from the inputs) is O(n^2). This is why the common answer is time O(n^3) and space O(n^2).