To solve the given problem, we need to find the inverse of a scaled matrix. The question asks for the inverse of the matrix \( (3A)^{-1} \) where \( A \) is a square matrix. Let's break down the steps:
- Understand the scaling property of matrix inverses:
The inverse of a scalar multiple of a matrix can be expressed as: \((kA)^{-1} = \frac{1}{k} A^{-1}\), where \( k \) is a non-zero scalar. - Apply the property to the given matrix:
Here, \( k = 3 \), and we want to find \( (3A)^{-1} \). According to the formula, \((3A)^{-1} = \frac{1}{3} A^{-1}\). - Verify the reasoning:
Using the properties of matrix operations, multiplying both sides of \( (3A)(3A)^{-1} = I \), we expect the expression \((3A)^{-1}\) to correctly equal the inverse scaled by the reciprocal of the scalar. - Conclusion:
The correct expression for \( (3A)^{-1} \) is \(\frac{1}{3} A^{-1}\).
With this analysis and calculation, option \(\dfrac{1}{3}A^{-1}\) is indeed the correct answer.
Let's eliminate other options:
- \(3A^{-1}\): This does not account for the reciprocal scaling factor when taking inverses of a scaled matrix.
- \(9A^{-1}\): Incorrect due to incorrect scaling calculation; no mathematical basis in this scalar manipulation.
- \(\frac{1}{9}A^{-1}\): Incorrect due to overestimating the reciprocal scaling effect.
Thus, the correct answer is the inverse scaled by the reciprocal \( \frac{1}{3} \), leading us to conclude the answer as \(\dfrac{1}{3}A^{-1}\).