Correct : b,c,d

1. Structure of the Matrix System: • Matrix A has dimensions n × m (n rows and m columns).
• The variable vector x has a dimension of m × 1 (representing m variables).
• We are given that m > n, meaning there are more variables than equations (a "short and wide" matrix).
2. Determine the Dimension of the Solution Space (Null Space): • Let r be the rank of matrix A. By definition, the rank cannot exceed the number of rows or columns:
    r ≤ n < m
• According to the Rank-Nullity Theorem, the dimension of the solution space (the nullity of A) is given by:
    Nullity = Number of columns - Rank = m - r
• Since the maximum possible rank r is n, the minimum dimension of the null space is:
    Nullity ≥ m - n
This means there are at least m - n linearly independent basis vectors that span the entire solution space. Every possible solution to Ax = 0 can be written as a linear combination of these basis vectors.
→ This validates Option (b).
3. Evaluate Particular Solution Conditions: • Option (c) - Non-zero solution with at least m - n zeros:
We can choose any m - n variables corresponding to free variables and fix them to 0. This leaves us with a reduced system of n equations and n variables. Since a homogeneous system with matching counts always has a solution, we can find a non-zero vector satisfying the original constraints.
→ This statement is TRUE.

• Option (d) - A solution with at least n non-zero variables:
Since there are at least m - n free variables, we can easily assign non-zero values to the free variables, which propagates non-zero values to the remaining dependent variables, yielding solutions where n or more elements are non-zero.
→ This statement is TRUE.
Conclusion: Statements b, c, and d are all theoretically true characteristics of the system.