If the equation $AX = b$ has no solution, it implies that the matrix $A$ is singular. A matrix is singular if it does not have an inverse, which occurs if and only if its determinant is zero.

Reasoning:

• For the equation $AX = b$ to have a solution, the matrix $A$ must be invertible, meaning there exists a unique solution $X = A^{-1}b$ when the determinant $\text{det}(A) \neq 0$.
• If $\text{det}(A) = 0$, the matrix $A$ is singular, meaning it has no inverse. In this case, $AX = b$ might have either no solution or infinitely many solutions depending on $b$.

Given that the problem specifies that $AX = b$ has no solution, this is a clear indication that the matrix $A$ is singular, and therefore:

$\text{det}(A) = 0$

Conclusion:

The correct answer is:

a. 0

Would you like more details, or do you have any questions?

Here are some related questions:

1. What does it mean for a matrix to be singular?
2. How can you determine if a matrix has no inverse?
3. What is the relationship between the determinant and the invertibility of a matrix?
4. How does the determinant relate to the solution of a system of linear equations?
5. What are the implications of $\text{det}(A) = 0$ for the solutions of $AX = b$?

Tip: A determinant of zero indicates that the matrix is singular, meaning the rows or columns are linearly dependent.