Are all slope coefficients correlated with the intercept in multiple linear regression? So in a simple linear regression

Y = B0 + B1X

B0 and B1 are negatively correlated.  When slope goes up, intercept must go down, in order for the regression line to still pass through as much of the data as possible.
My question is, does the same intuition apply for a multiple linear regression assuming all the predictors are independent from each other.  Would each slope coefficient, Bi, individually be negatively correlated with intercept, B0?
Thanks!
 A: Correlation of the estimated coefficients in linear regression: For linear regression using OLS estimation of the coefficients, the general formula for the variance of the estimated coefficient vector (implicitly conditional on the explanatory variables) is:
$$\mathbb{V}(\hat{\boldsymbol{\beta}}) 
= \sigma^2 (\mathbf{x}^\text{T} \mathbf{x})^{-1} 
= \sigma^2 \cdot \frac{\text{adj} (\mathbf{x}^\text{T} \mathbf{x})}{\text{det} (\mathbf{x}^\text{T} \mathbf{x})}.$$
The resultant correlation between the estimated intercept and the $k$th estimated slope parameter can be written in terms of the adjugate matrix of the Gramian of the design matrix, as follows:
$$\mathbb{Corr}(\hat{\beta}_0,\hat{\beta}_k) = \frac{[\text{adj} (\mathbf{x}^\text{T} \mathbf{x})]_{0,k}}{\sqrt{[\text{adj} (\mathbf{x}^\text{T} \mathbf{x})]_{0,0} \cdot [\text{adj} (\mathbf{x}^\text{T} \mathbf{x})]_{k,k}}}.$$
(Note that in this formula label the rows and columns as $k=0,1,...,m$, so the element $k$ refers to the $k$th slope coefficient.)  This is a general formula for the correlation for a linear regression model with any number of coefficients.  In the case where you have a large number of coefficients, the form of the adjugate matrix will be complicated, and will involve a substantial amount of interaction between the explanatory variables.

Correlation of the estimated coefficients in simple linear regression: For the case of simple linear regression this simplifies to:
$$\mathbb{V}(\hat{\boldsymbol{\beta}}) 
= \frac{1}{n(\sum x_i^2) - (\sum x_i)^2} \begin{bmatrix} 
\sum x_i^2 & - \sum x_i \\
- \sum x_i & n \\
 \end{bmatrix},$$
and so, letting $\bar{x} = (\sum x_i)/n$ and $\bar{\bar{x}} = (\sum x_i^2)/n$, we have:
$$\mathbb{Corr}(\hat{\beta}_0,\hat{\beta}_1) = - \frac{\sum x_i}{\sqrt{n \sum x_i^2}}
= - \frac{\bar{x}}{\sqrt{\bar{\bar{x}}}}.$$
Thus, in the case of simple linear regression, the correlation between the estimated slope and the estimated intercept has opposite sign to the mean of the explanatory variables.  If the mean of the explanatory variables is positive, the correlation between the estimated coefficients will be negative.  If the mean of the explanatory variables is negative, the correlation between the estimated coefficients will be positive.  If the mean of the explanatory variables is zero, the correlation between the estimated coefficients will also be zero.
