For an 84×102 matrix A, the rank cannot be greater than 84, which is the number of rows, and the nullity cannot be less than 18, which is the number of columns minus the rank. If the rank is 47, the solutions of Ax=b will have 55 free variables and the nullity of A is 55.
The question regarding a 84×102 matrix A pertains to linear algebra and involves understanding the concepts of matrix rank and nullity. The rank of a matrix is the maximum number of linearly independent rows or columns. For matrix A, its rank cannot exceed 84, as that is the smaller dimension of the matrix. Consequently, the nullity of A, which is defined as the dimension of the null space of A, is given by the formula nullity(A) = number of columns - rank(A). Thus, the nullity of A cannot be less than 18, which is 102 (number of columns) minus 84 (maximum possible rank).
For part (b), if the rank of A is 47, the nullity is thus 102 - 47 = 55. This implies that the solutions to the matrix equation Ax = b will have 55 free variables because each free variable corresponds to a dimension in the null space of A. This situation is typical in underdetermined systems, where more variables exist than equations.