If the length of a wire is made double and radius is halved of its respective values. Then, the Young's modulus of the material of the wire will:

Young's modulus is a material property that depends only on the nature of the material, not on the dimensions of the specimen. It is defined as $$Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L}$$, where $$F$$ is the applied force, $$A$$ is the cross-sectional area, $$\Delta L$$ is the change in length, and $$L$$ is the original length.

When the length is doubled and the radius is halved, these are changes to the geometry of the wire, not to its material composition. Since Young's modulus is an intrinsic property of the material, it does not change when the dimensions of the wire are altered.

Hence, the correct answer is Option A.