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# numpy.linalg.solve

`numpy.linalg.``solve`(a, b)

  Solve a linear matrix equation, or system of linear scalar equations.

  Computes the “exact” solution, x, of the well-determined, i.e., full rank, linear matrix equation ax = b.

  Parameters : | a : array_like, shape (M, M) Coefficient matrix. b : array_like, shape (M,) or (M, N) Ordinate or “dependent variable” values.
  Returns : | x : ndarray, shape (M,) or (M, N) depending on b Solution to the system a x = b
  Raises : | LinAlgError : If a is singular or not square.

  Notes

  solve is a wrapper for the LAPACK routines dgesv and zgesv, the former being used if a is real-valued, the latter if it is complex-valued. The solution to the system of linear equations is computed using an LU decomposition [R40] with partial pivoting and row interchanges.

  a must be square and of full-rank, i.e., all rows (or, equivalently, columns) must be linearly independent; if either is not true, use lstsq for the least-squares best “solution” of the system/equation.

  References

  [R40] | (1, 2) G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 22.

  Examples

  Solve the system of equations `3 * x0 + x1 = 9` and `x0 + 2 * x1 = 8`:

      >>> a = np.array([[3,1], [1,2]])
      >>> b = np.array([9,8])
      >>> x = np.linalg.solve(a, b)
      >>> x
      array([ 2.,  3.])

  Check that the solution is correct:

      >>> (np.dot(a, x) == b).all()
      True

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