Question

if a is a 3 ✕ 5 matrix, what are the possible values of nullity(a)? (enter your answers as a comma-separated list.)

Answer 1

The possible values of nullity(a) for a 3 ✕ 5 matrix range from 0 to 5, inclusive, depending on the rank of the matrix.

Step-by-step explanation:

The nullity of a matrix is the dimension of its null space, which is also known as the kernel. The null space consists of all vectors that, when multiplied by the matrix, result in the zero vector. To determine the possible values of the nullity of a 3 ✕ 5 matrix, we need to analyze its rank. The rank of a matrix is the maximum number of linearly independent rows or columns it contains.

Since the matrix has 3 rows and 5 columns, its rank cannot exceed 3. Therefore, the possible values of nullity(a) can range from 0 to 5, inclusive, depending on the rank of the matrix. If the matrix has full rank (3), the nullity would be 0. If the rank is less than 3, the nullity would be the difference between 5 and the rank.

Answer 2

The possible values of the nullity of a 3 × 5 matrix A are 2, 3, 4, and 5. This conclusion is based on the Rank-Nullity Theorem and the fact that the rank of the matrix cannot exceed the number of its rows, which is 3 in this case.

Step-by-step explanation:

Given that A is a 3 × 5 matrix, the possible values of the nullity of A relate to the dimension of the null space of A. The Rank-Nullity Theorem states that for any matrix A, the rank of A plus the nullity of A is equal to the number of columns in A. Since A has 5 columns, if we let r be the rank of A, then we have r + nullity(A) = 5. The rank r cannot exceed the number of rows, which is 3, and because the rank is a non-negative integer, r can range from 0 to 3. Therefore, the possible values of the nullity of A can be obtained by subtracting the rank from 5, giving us the possible nullity values of 2, 3, 4, and 5 when the rank is respectively 3, 2, 1, and 0.