Final answer:
The nullity of a matrix is the dimension of its null space, and according to the Rank-Nullity Theorem, the nullity plus the rank equals the number of columns in the matrix. The biggest possible value for the nullity of a matrix is thus limited by the size of the matrix. Option d) It depends on the size of matrix is the correct answer.
Step-by-step explanation:
The question asks about the biggest possible value for the nullity of a matrix A. The nullity of a matrix is the dimension of the null space of the matrix, which is the set of all solutions to the homogeneous equation Ax = 0, where A is the matrix and x is a column vector. The null space (also known as the kernel) represents all the vectors that are mapped to the zero vector when multiplied by the matrix A.
In linear algebra, there is an important theorem called the Rank-Nullity Theorem, which states that the rank of a matrix plus its nullity is equal to the number of columns in the matrix. This theorem constraints the nullity of the matrix based on its size. Given this theorem, the nullity cannot be arbitrarily large and is constrained by the size of the matrix. Therefore, if we have an n × n matrix, the maximum nullity it can have is n, which occurs when the rank is 0.
To conclude, the biggest possible value for the nullity of a matrix is constrained by the size of the matrix, specifically the number of columns. Thus, the correct option in this case is (d) It depends on the size of matrix A.