Question

Let A be square matrix of order m, then nullity of A is

• A. \( 2 \cdot \text{Rank of } [A] - m \)
• B. \( m + \text{Rank of } [A] \)
• C. \( m \)
• D. \( m - \text{Rank of } [A] \)

Hint:

Use the rank-nullity theorem to easily calculate the nullity of a matrix. The rank plus the nullity always equals the order of the matrix.

Answer 1

The Correct Option is D

The rank-nullity theorem relates the rank and nullity of a matrix \( A \). For a square matrix \( A \) of order \( m \), the rank-nullity theorem states: \[ \text{Rank of } A + \text{Nullity of } A = m \] The nullity of matrix \( A \) is defined as the dimension of the null space of \( A \), which is the number of linearly independent solutions to the equation \( A \mathbf{x} = 0 \).
Thus, the nullity of \( A \) is given by:
\[ \text{Nullity of } A = m - \text{Rank of } A \] This is the correct expression for the nullity of a square matrix of order \( m \).

(1) Why Other Options Are Incorrect:
- Option 1: \( 2 \cdot \text{Rank of } [A] - m \) is not correct, as it does not correspond to the rank-nullity theorem.
- Option 2: \( m + \text{Rank of } [A] \) is incorrect as it does not give the correct relationship between rank and nullity.
- Option 3: \( m \) is incorrect as it represents the order of the matrix, not the nullity.

Conclusion: The nullity of matrix \( A \) is \( m - \text{Rank of } A \).