The dimension of the modulus of rigidity, is

The correct option is [ML⁻¹T⁻²].

The modulus of rigidity (also known as the shear modulus) is defined as the ratio of shear stress to shear strain:
$\mathrm{Modulus\; of\; Rigidity}=\frac{\mathrm{Shear\; Stress}}{\mathrm{Shear\; Strain}}$

First, we find the dimensions of shear stress. Shear stress is defined as force per unit area:
$\mathrm{Shear\; Stress}=\frac{\mathrm{Force}}{\mathrm{Area}}$
The dimensional formula for force is:
$\left[\mathrm{Force}\right]=\left[ML{T}^{-2}\right]$
The dimensional formula for area is:
$\left[\mathrm{Area}\right]=\left[L^2\right]$
Therefore, the dimensional formula for shear stress is:
$\left[\mathrm{Shear\; Stress}\right]=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}=\left[M{L}^{-1}{T}^{-2}\right]$

Next, shear strain is defined as a ratio of change in dimension to the original dimension, which means it is a dimensionless quantity:
$\left[\mathrm{Shear\; Strain}\right]=\left[M^0{L}^0{T}^0\right]$

Substituting the dimensions of shear stress and shear strain back into the formula for the modulus of rigidity:
$\left[\mathrm{Modulus\; of\; Rigidity}\right]=\frac{\left[M{L}^{-1}{T}^{-2}\right]}{\left[M^0{L}^0{T}^0\right]}=\left[M{L}^{-1}{T}^{-2}\right]$
Thus, the dimensional formula of the modulus of rigidity is [ML⁻¹T⁻²].