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!