We are given that $M$ is a square matrix with a zero determinant, and we are asked to determine which of the statements (S1–S4) are correct. Let's analyze each one.

1. S1: Each row of $M$ can be represented as a linear combination of the other rows

   • If the determinant of a matrix is zero, the rows of the matrix are linearly dependent. This means at least one row can be written as a linear combination of the other rows. So, S1 is correct.

2. S2: Each column of $M$ can be represented as a linear combination of the other columns

   • Similarly, if the determinant is zero, the columns of the matrix are linearly dependent as well. This implies that at least one column can be represented as a linear combination of the other columns. Therefore, S2 is correct.

3. S3: $M X = 0$ has a nontrivial solution

   • When a matrix has a zero determinant, it is singular. For a singular matrix, the homogeneous equation $MX = 0$ has nontrivial solutions. This is because the rank of the matrix is less than the number of unknowns, leading to free variables in the system. Thus, S3 is correct.

4. S4: $M$ has an inverse

   • A matrix with a zero determinant does not have an inverse. This is a basic property of singular matrices. Hence, S4 is incorrect.

Conclusion:

The correct statements are S1, S2, and S3.

Would you like further details or have any questions?

Here are 5 related questions to further your understanding:

1. What are the properties of a matrix with a zero determinant?
2. Can a non-square matrix have a determinant? Why or why not?
3. How do you calculate the inverse of a matrix if it exists?
4. What is the rank of a matrix, and how does it relate to linear dependence?
5. Why do singular matrices not have an inverse?

Tip: A matrix with a zero determinant is always singular and cannot have an inverse.