Static 1D Fusion: Optimal Inverse-Variance Weighting and Fused Variance

Two independent, unbiased sensors each measure the same constant scalar quantity (e.g., the distance to a wall). Sensor A reports $z_A = 10.0$ with variance $\sigma_A^2 = 4.0\ \mathrm{m}^2$. Sensor B reports $z_B = 11.0$ with variance $\sigma_B^2 = 1.0\ \mathrm{m}^2$. Using the optimal (minimum-variance / maximum-likelihood, inverse-variance-weighted) static fusion of two independent Gaussian estimates, compute the variance of the fused estimate. Give the answer in $\mathrm{m}^2$.

For reference, the fused mean is $\hat{z} = \dfrac{\sigma_B^2\,z_A + \sigma_A^2\,z_B}{\sigma_A^2 + \sigma_B^2}$ and the fused variance satisfies $\dfrac{1}{\sigma^2} = \dfrac{1}{\sigma_A^2} + \dfrac{1}{\sigma_B^2}$.