https://file.aiquickdraw.com/mathbot/file/17273523570206zskkxv6.jpg

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.

If M is a square matrix with a zero determinant, which of the following assertions are correct?

Analyze the properties of a square matrix with a zero determinant, including linear dependence of rows and columns, nontrivial solutions for MX=0, and the lack of an inverse. This problem covers key concepts in linear algebra and matrix theory, suitable for advanced high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation