Question

Consider a linear homogeneous system of equations \( A\mathbf{x} = \mathbf{0} \), where \( A \) is an \( n \times n \) matrix, \( \mathbf{x} \) is an \( n \times 1 \) vector, and \( \mathbf{0} \) is an \( n \times 1 \) null vector. Let \( r \) be the rank of \( A \). For a non-trivial solution to exist, which of the following conditions is/are satisfied?

• A. Determinant of \( A = 0 \)
• B. \( r = m<n \)
• C. \( r<n \)
• D. Determinant of \( A \neq 0 \)

Hint:

For homogeneous systems, the determinant and rank conditions determine the existence of non-trivial solutions. A singular matrix (\( \det(A) = 0 \)) and rank \( r<n \) are necessary for non-trivial solutions.

Answer 1

The Correct Option is A

Step 1: Analyze the system of equations \( A\mathbf{x} = \mathbf{0} \). A non-trivial solution to a homogeneous system \( A\mathbf{x} = \mathbf{0} \) exists only if the determinant of \( A \) is zero. This is because:
1. If \( \det(A) \neq 0 \), \( A \) is invertible, and the only solution is the trivial solution \( \mathbf{x} = \mathbf{0} \).
2. If \( \det(A) = 0 \), \( A \) is singular, and there are infinitely many solutions, including non-trivial solutions.
Step 2: Rank condition. The rank \( r \) of \( A \) determines the number of independent rows (or columns) of the matrix. For \( A\mathbf{x} = \mathbf{0} \):
1. If \( r = n \), \( \mathbf{x} = \mathbf{0} \) is the only solution (trivial solution).
2. If \( r<n \), there are \( n - r \) free variables, and a non-trivial solution exists.
Step 3: Evaluate the conditions.
- Condition (A): Determinant of \( A = 0 \) is satisfied for a non-trivial solution.
- Condition (B): \( r = m<n \) is not satisfied because \( r<n \) ensures a non-trivial solution.
- Condition (C): \( r<n \) is satisfied as it directly implies a non-trivial solution.
- Condition (D): Determinant of \( A \neq 0 \) contradicts the requirement for a non-trivial solution.
Step 4: Conclusion. The conditions satisfied for a non-trivial solution are: \[ \text{(A) Determinant of \( A = 0 \), (C) \( r<n \)}. \]