The formula for mutual inductance is given by, $\epsilon =M\frac{dI}{dt} \Rightarrow M=\frac{\epsilon }{\left({\displaystyle \frac{dI}{dt}}\right)}$

Dimension of emf or voltage is $\left[\mathrm{M} {\mathrm{L}}^2 {\mathrm{T}}^{-3} {\mathrm{A}}^{-1}\right]$,
Dimension of current is $\left[\mathrm{A}\right]$,
Dimension of time $\left[\mathrm{T}\right]$.

Therefore, the dimension of $M$ is
$M=\frac{\left[{\mathrm{ML}}^2{\mathrm{T}}^{-3}{\mathrm{A}}^{-1}\right]}{\left[\mathrm{A}\right]{\left[\mathrm{T}\right]}^{-1}}$
$=\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}{\mathrm{A}}^{-2}\right]$