The question is asking about non-singular matrices and which of the given statements are true. Let's go through each one.

1. In a non-singular matrix, rows are linearly dependent.

   • False. In a non-singular matrix, the rows are linearly independent, which means no row can be written as a linear combination of the others.

2. In a non-singular matrix, rows are linearly independent.

   • True. By definition, a non-singular matrix has linearly independent rows. This is why its determinant is non-zero and why it has an inverse.

3. In a non-singular matrix there is only a unique solution for the represented system of equations.

   • True. A non-singular matrix implies that the corresponding system of linear equations has a unique solution because its determinant is non-zero, making the system solvable.

4. In a non-singular matrix, one row can be a multiple of another one.

   • False. If one row were a multiple of another, the rows would be linearly dependent, which contradicts the definition of a non-singular matrix.

Correct answers:

• The second and third options are correct:
  • In a non-singular matrix, rows are linearly independent.
  • In a non-singular matrix there is only a unique solution for the represented system of equations.

Would you like more details on this? Here are some related questions to further explore this topic:

1. What is the relationship between the determinant of a matrix and its singularity?
2. How can you determine if a matrix is non-singular using Gaussian elimination?
3. What role do linearly independent vectors play in defining the rank of a matrix?
4. Can a matrix with zero rows or columns ever be non-singular?
5. How does the inverse of a matrix relate to its non-singularity?

Tip: A matrix is singular if its determinant is zero, meaning it does not have an inverse, and its rows or columns are linearly dependent.