## associativity of matrix multiplication in numpy

Question

I am playing with a simple numpy example and having hard time to understand why associative property of matrix multiplication

```
ABC = (AB)C = A(BC)
```

does not exactly hold. I assume the problem is with numeric stability. But how to address it? What is the issue exactly?

Here is my example with linear regression. I use sklearn solution as it gives more divergence between associative groupings:

```
import numpy as np
np.random.seed(42)
num_samples = 100
M = 1000
sigma = 0.5
X = np.random.binomial(2, 0.4, (num_samples, M))
beta = np.zeros(M)
beta[5] = 1.0
y = X.dot(beta) + sigma*np.random.randn(num_samples)
"standardise y"
y = y - np.mean(y)
y = y/np.std(y)
"center and standardise X"
Xc = X - X.mean(axis=0)
xstd = X.std(axis=0)
mask = xstd > 1e-12
Xc = Xc[:, mask]
from sklearn.linear_model import LinearRegression
lr = LinearRegression()
lr.fit(Xc ,y)
beta_hat_sklearn = lr.coef_
beta_hat_sklearn.T @ Xc.T @ Xc @ beta_hat_sklearn / num_samples
"equivalent < Python3.5"
beta_hat_sklearn.T.dot(Xc.T).dot(Xc).dot(beta_hat_sklearn) / num_samples
# 1.0000000000000009
beta_hat_sklearn.T @ (Xc.T @ Xc) @ beta_hat_sklearn / num_samples
"equivalent < Python3.5"
beta_hat_sklearn.T.dot(Xc.T.dot( Xc )).dot(beta_hat_sklearn )/ num_samples
# 0.89517439485479278
```

### Update

It might be MacOSX specific bug.

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