A square matrix has inverse, if `|A|` is

Options

A. 0
B. 1
C. `-1`
D. non zero

Answer

To determine the condition under which a square matrix \( A \) has an inverse, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Inverse of a Matrix**:
The inverse of a matrix \( A \) is denoted as \( A^{-1} \). A matrix has an inverse if and only if it is non-singular.

2. **Condition for Invertibility**:
For a square matrix \( A \) to have an inverse, the determinant of \( A \), denoted as \( |A| \), must be non-zero. This is because the formula for the inverse of a matrix involves the determinant:
\[ A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A) \]
where \( \text{adj}(A) \) is the adjoint of \( A \).

3. **Conclusion**:
Therefore, the condition for a square matrix \( A \) to have an inverse is:
\[ |A| \neq 0 \]

### Final Answer:
A square matrix has an inverse if \( |A| \) is **non-zero**.

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