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Hint: In this question we will use the property of matrices according to the question to solve the given problem.

Complete step-by-step answer:

Now, according to question A is involuntary matrix which means ${{\text{A}}^2} = {\text{I}}$ where ${\text{I}}$ is an Identity matrix which is a square matrix. All the diagonal elements of the identity matrix have value equal to 1. Except diagonal elements all other elements have value which is equal to 0. Now, using the property ${{\text{A}}^2} = {\text{I}}$, we get

${{\text{A}}^2} = {\text{I}}$ $ \Rightarrow $ ${\text{AA}} = {\text{I}}$ $ \Rightarrow $ ${\text{A = }}{{\text{A}}^{ - 1}}$ where ${{\text{A}}^{ - 1}}$ is the inverse of A.

Now, ${\text{AA}} = {\text{I}}$ …… (1)

Multiply and divide the left-hand side by 2, we get

$\dfrac{{\text{A}}}{2}(2{\text{A) = I}}$ , where 2A is the inverse of $\dfrac{{\text{A}}}{2}$.

So, the answer is option (A) i.e. 2A.

Note: In such types of problems most of the students start finding the inverse asked in the question by applying the longer method i.e. by finding the adjoint and modulus of the matrix which is a very tedious process. Such questions are easy and are solved by just applying the property. We can solve them in just a few lines.

Complete step-by-step answer:

Now, according to question A is involuntary matrix which means ${{\text{A}}^2} = {\text{I}}$ where ${\text{I}}$ is an Identity matrix which is a square matrix. All the diagonal elements of the identity matrix have value equal to 1. Except diagonal elements all other elements have value which is equal to 0. Now, using the property ${{\text{A}}^2} = {\text{I}}$, we get

${{\text{A}}^2} = {\text{I}}$ $ \Rightarrow $ ${\text{AA}} = {\text{I}}$ $ \Rightarrow $ ${\text{A = }}{{\text{A}}^{ - 1}}$ where ${{\text{A}}^{ - 1}}$ is the inverse of A.

Now, ${\text{AA}} = {\text{I}}$ …… (1)

Multiply and divide the left-hand side by 2, we get

$\dfrac{{\text{A}}}{2}(2{\text{A) = I}}$ , where 2A is the inverse of $\dfrac{{\text{A}}}{2}$.

So, the answer is option (A) i.e. 2A.

Note: In such types of problems most of the students start finding the inverse asked in the question by applying the longer method i.e. by finding the adjoint and modulus of the matrix which is a very tedious process. Such questions are easy and are solved by just applying the property. We can solve them in just a few lines.

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