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The problem asks which of the following conditions imply that the determinant of a square matrix \( A \) is zero.

The statements are:

a. \( A \) is equal to its transpose.  
b. All the entries of \( A \) are negative.  
c. The matrix \( A \) is not invertible.  
d. Two columns of \( A \) are equal.

### Analysis:
- **Option a: \( A \) is equal to its transpose**  
  This means \( A \) is symmetric. A symmetric matrix can still have a nonzero determinant, so this option does *not* necessarily imply that \( \det(A) = 0 \).

- **Option b: All entries of \( A \) are negative**  
  The signs of the entries do not directly determine whether the determinant is zero. So, this does *not* imply \( \det(A) = 0 \).

- **Option c: \( A \) is not invertible**  
  A matrix is invertible if and only if its determinant is nonzero. If \( A \) is not invertible, it means \( \det(A) = 0 \). This statement is **true**.

- **Option d: Two columns of \( A \) are equal**  
  If two columns of a matrix are equal, the matrix is singular, meaning it has a determinant of zero. So, this option is **true**.

### Correct answers:
- **c. The matrix \( A \) is not invertible**
- **d. Two columns of \( A \) are equal**

These two conditions imply that \( \det(A) = 0 \).

Do you want further details on any of these options? 

Here are some related questions to explore:

1. What are the properties of an invertible matrix?
2. How does the determinant relate to linear independence of columns?
3. What is the geometric interpretation of a matrix having a determinant of zero?
4. Can a matrix with all positive or all negative entries have a zero determinant?
5. How do elementary row operations affect the determinant of a matrix?

**Tip:** A matrix with linearly dependent rows or columns always has a determinant of zero!

Let A be a square matrix. Which statements from the following list imply that det(A) = 0?

Explore the conditions that imply the determinant of a square matrix A equals zero. Learn about matrix invertibility, singular matrices, and the effects of column dependencies on the determinant.

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