# Conjugate inverseGamma prior and Multivariate normal?

I do know that inverse gamma is a conjugate prior for univariate normal distribution. I guess it's also a conjugate prior for multivariate normal distribution. I'm trying to get the closed-form posterior from an inverse gamma prior and a likelihood based on a multivariate normal distribution expecting to get inverse gamma posterior but I don't have any success yet. I appreciate it if you could help me get the posterior. Below is the model:

$$Y_i | \kappa_i \sim \mathcal{N}_j(0, \kappa_i^2 1 1^t + \sigma^2 I_j)$$ $$\kappa_i \sim \text{InverseGamma} (A, B)$$

Where $A$, $B$, and $\sigma^2$ are known constants. $1$ is a vector of 1 of length $j$ and $I$ is a $j$ by $j$ identity matrix.