Given a 3×3
matrix A
 and a vector b∈R3
. If the equation AX=b
has no solution, then det(A)=?
Question 7Answer

a.
0

b.
1

c.
2

d.
3

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.

Explore the relationship between determinants and the solutions of linear equations AX=b. Learn why det(A)=0 indicates no solution for AX=b and how singular matrices affect solutions. Detailed explanation suitable for college-level students.

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