Question

In each case below, determine the largest possible value for rank (A) and the smallest possible value for nullity(A):

1.) A is 3x3

2.) A is 3x4

3.) A is 5x4

4.) A is 3x5

Answer 1

The problem involves determining the maximum rank and minimum nullity for matrices of various sizes using the rank-nullity theorem. The rank of a matrix is the dimension of its column space, while nullity is the dimension of its null space. The solutions utilize the theorem that rank(A) plus nullity(A) equals to the number of columns in A.

Step-by-step explanation:

The question posed is related to linear algebra, specifically finding the maximal rank and minimal nullity of a matrix A of different sizes. The rank of a matrix A is the dimension of the column space of A, which is the same as the dimension of the row space of A. The nullity of a matrix A is the dimension of the null space of A. According to the rank-nullity theorem, for a matrix A of size m × n, rank(A) + nullity(A) = n.

1. For a 3x3 matrix, the largest rank is 3 and the smallest nullity is 0.
2. For a 3x4 matrix, the largest rank is 3 because there are only 3 rows, and the smallest nullity is 1 since rank + nullity = 4.
3. For a 5x4 matrix, the largest rank is 4 due to a maximum of 4 linearly independent columns, and the smallest nullity is 0.
4. For a 3x5 matrix, the largest rank is 3, and the smallest nullity is 2 because rank + nullity must equal the number of columns, which is 5.