A matrix has a non-trivial solution when it forms a system of linear equations that has infinitely many solutions. This typically occurs when the matrix is inconsistent or has a determinant of zero, indicating that the equations represented by the matrix are dependent.
In mathematical terms, if we consider a homogeneous system of equations represented in matrix form as Ax = 0, where A is a matrix and x is a vector of variables, a non-trivial solution refers to any solution where x is not equal to the zero vector.
For a non-trivial solution to exist, the following conditions must be met:
- The number of variables must exceed the number of independent equations in the system.
- The determinant of the matrix must be zero, which indicates that the rows (or columns) of the matrix are linearly dependent.
In summary, if the matrix does not have full rank and the number of variables is greater than the number of linearly independent equations, a non-trivial solution is possible.