A square matrix A has inverse if and only if A is ............

Answer

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

### Step 1: Understanding the Inverse of a Matrix
A square matrix \( A \) has an inverse, denoted as \( A^{-1} \), if there exists another matrix such that when multiplied together, they yield the identity matrix \( I \). This means:
\[ A \cdot A^{-1} = I \]

### Step 2: Condition for the Existence of Inverse
For a square matrix \( A \) to have an inverse, it must be non-singular. This means that the determinant of \( A \), denoted as \( \text{det}(A) \), must not be equal to zero:
\[ \text{det}(A) \neq 0 \]

### Step 3: Relationship Between Determinant and Inverse
The formula for the inverse of a matrix \( A \) is given by:
\[ A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)} \]
where \( \text{adj}(A) \) is the adjugate of \( A \). If \( \text{det}(A) = 0 \), then the expression for \( A^{-1} \) becomes undefined because division by zero is not possible.

### Step 4: Conclusion
Therefore, a square matrix \( A \) has an inverse if and only if it is non-singular, which is equivalent to saying that:
\[ \text{det}(A) \neq 0 \]

### Final Answer
A square matrix \( A \) has an inverse if and only if \( A \) is **non-singular**.

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