Exercise 4.3.14

The matrix $(\hat{x}-x){(\hat{x}-x)}^T$ is ${\left(A^{T}A\right)}^{-1}{A}^T(b-Ax){(b-Ax)}^{T}A{\left(A^{T}A\right)}^{-1}$. When the average of $(b-Ax){(b-Ax)}^T$ is ${\sigma }^{2}I$, the average of $(\hat{x}-x){(\hat{x}-x)}^T$ will be the output covariance matrix ${\left(A^{T}A\right)}^{-1}{A}^T{\sigma }^{2}A{\left(A^{T}A\right)}^{-1}$ which simplifies to ${\sigma }^2{\left(A^{T}A\right)}^{-1}$. That gives the average of the squared output errors $\hat{x}-x$.