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The question on the screen asks which statements imply that the determinant of matrix \( A \), denoted as \( \det(A) \), is equal to 0. Let’s go through each option:

1. **a. All the entries of \( A \) are negative.**
   - This does not necessarily imply that \( \det(A) = 0 \). The determinant is not directly related to the sign of the entries. The entries can be negative, but the matrix could still be invertible and have a non-zero determinant.

2. **b. \( A \) is equal to its transpose.**
   - This means \( A \) is symmetric. Symmetry alone does not imply that the determinant is zero. A symmetric matrix can have a non-zero determinant.

3. **c. The matrix \( A \) is not invertible.**
   - This statement directly implies that \( \det(A) = 0 \). If a matrix is not invertible, then by definition, its determinant must be zero.

4. **d. Two columns of \( A \) are equal.**
   - If two columns of a matrix are identical, the matrix is linearly dependent, which causes the determinant to be zero.

Therefore, the correct answers are **c** and **d**, as both conditions imply that \( \det(A) = 0 \).

Would you like more details or have any questions?

Here are some related questions to explore further:

1. What other conditions can lead to a matrix having a zero determinant?
2. How can we determine if a matrix is invertible using its determinant?
3. What is the geometric interpretation of a zero determinant in a matrix?
4. How does swapping two rows or columns in a matrix affect its determinant?
5. Can a matrix with all non-zero entries still have a zero determinant?

**Tip:** Remember that if a matrix has linearly dependent rows or columns, its determinant will be zero.

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

Explore the conditions under which the determinant of a square matrix is zero. Learn how matrix properties like invertibility and column linear dependence influence the determinant. Suitable for undergraduate students studying linear algebra.

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