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.