The problem involves a homogeneous system of equations given by \(AX=O\), where \(A\) is a matrix, and \(X = \begin{pmatrix}l \\ m \\ 0 \end{pmatrix}\) is a solution vector. We know that this vector represents an infinite number of solutions and the aim is to determine the rank of matrix \(A\).
- Firstly, recall that if a homogeneous system of linear equations has a non-trivial solution (other than the trivial solution \(X=0\)), then the determinant of the matrix of coefficients (\(A\)) must be zero. This implies that the rows of \(A\) are linearly dependent, and hence not all rows can be independent.
- In this specific problem, the solution vector is \(X = \begin{pmatrix}l \\ m \\ 0 \end{pmatrix}\), where \(l \neq 0\) and \(m \neq 0\). The vector suggests there is at least one non-zero solution.
- For a system of three linear equations with three unknowns to have infinite solutions with at least one parameterized dimension, the rank of matrix \(A\) should be less than the number of unknowns. Hence, the rank can't be 3 because that would only have the trivial solution, not the infinite solutions.
- If the rank were 2, it would imply dependency between only one of the rows, providing a set parameterization, while the vector given, \(X = \begin{pmatrix}l \\ m \\ 0 \end{pmatrix}\), suggests a more pronounced indeterminacy indicative of even lower independence.
- Thus, the rank should be 1, indicating strong linear dependency among the rows of \(A\), leading to two free parameters capable of taking any real value, except jointly zero.
Therefore, the rank of matrix \(A\) is 1.