If A is a `3xx3` matrix such that det.A=0, then

Options

A. A=0
B. A is non-singular
C. all elements of A are equal
D. A is singular

Answer

To solve the problem, we need to analyze the properties of a 3x3 matrix \( A \) given that its determinant \( \text{det}(A) = 0 \).

### Step-by-Step Solution:

1. **Understanding the Determinant**:
The determinant of a matrix provides important information about the matrix. If the determinant of a matrix is zero, it indicates that the matrix does not have full rank.

**Hint**: Recall that the determinant can tell us about the invertibility of a matrix.

2. **Definition of Singular Matrix**:
A matrix is called a singular matrix if its determinant is zero. This means that the matrix does not have an inverse.

**Hint**: Remember that a non-singular matrix has a non-zero determinant and is invertible.

3. **Conclusion about Matrix \( A \)**:
Since \( \text{det}(A) = 0 \), we conclude that matrix \( A \) is singular. Therefore, it does not have an inverse.

**Hint**: Think about the implications of a matrix being singular in terms of solutions to linear equations.

4. **Implications**:
If a matrix is singular, it may represent a system of linear equations that either has no solutions or an infinite number of solutions.

**Hint**: Consider how the rank of the matrix relates to the solutions of the system of equations it represents.

### Final Answer:
If \( A \) is a \( 3 \times 3 \) matrix such that \( \text{det}(A) = 0 \), then \( A \) is a singular matrix, and it does not have an inverse.