Question

How do you know how many fundamental solutions a matrix has based on the rank and dimensions?

A) Fundamental solutions equal the rank of the matrix
B) Fundamental solutions equal the difference between rank and dimensions
C) Fundamental solutions equal the dimensions of the matrix
D) Fundamental solutions equal the sum of rank and dimensions

Answer 1

The number of fundamental solutions of a matrix is equal to the difference between the number of dimensions (columns) and the rank of the matrix.

Step-by-step explanation:

The number of fundamental solutions of a matrix depends on its rank and its dimensions. Specifically, the number of fundamental solutions, also referred to as the nullity or dimension of the null space, is given by the difference between the number of columns (variables) and the rank of the matrix. Therefore, the correct answer to how you determine the number of fundamental solutions is B) Fundamental solutions equal the difference between the number of dimensions (typically the number of columns) and the rank of the matrix.

For example, if a matrix has 5 columns (variables) and a rank of 3, then it would have 5 - 3 = 2 fundamental solutions. This means there are two independent vectors that form the basis for the null space of the matrix.