Orthogonality & Least Squares · Superchallenge problem

Give the space of real $4\times 4$ matrices the Frobenius inner product $\langle X, Y\rangle = \sum_{i,j} x_{ij}y_{ij}$. Let $W$ be the subspace of matrices in which every row and every column sums to $0$. Find the distance from $A = \operatorname{diag}(1, 2, 3, 4)$ to $W$.