`(aA)^(-1)=1/aA^(-1)` where `a` is any real number and `A` is a square matrix.

Explanation

To determine whether the statement \((aA)^{-1} = \frac{1}{a} A^{-1}\) is true or false, where \(a\) is any real number and \(A\) is a square matrix, we will analyze both sides of the equation step by step.

### Step 1: Understand the Inverse of a Scalar-Matrix Product
The inverse of a product of a scalar and a matrix can be expressed using the properties of inverses. Specifically, if \(A\) is a non-singular (invertible) matrix and \(a \neq 0\), we can use the following property:
\[ (aA)^{-1} = \frac{1}{a} A^{-1} \]

### Step 2: Conditions for the Inverse to Exist
For the equation \((aA)^{-1} = \frac{1}{a} A^{-1}\) to hold true, two conditions must be satisfied:
1. The matrix \(A\) must be non-singular (invertible).
2. The scalar \(a\) must be non-zero (since division by zero is undefined).

### Step 3: Analyze the Case When \(a = 0\)
If \(a = 0\), then the expression \(aA\) becomes the zero matrix, which is singular. The inverse of the zero matrix does not exist. Therefore, in this case, the left-hand side \((0A)^{-1}\) is undefined, while the right-hand side \(\frac{1}{0} A^{-1}\) is also undefined.

### Step 4: Conclusion
Since the statement must hold for all real numbers \(a\), and we found that it does not hold when \(a = 0\), we conclude that the statement is false.

### Final Answer
The statement \((aA)^{-1} = \frac{1}{a} A^{-1}\) is **false**.

---