My MATLAB (R2010b) says quite a lot about what A\B does:
mldivide(A,B) and the equivalent A\B perform matrix left division
(back slash). A and B must be matrices that have the same number of
rows, unless A is a scalar, in which case A\B performs element-wise
division — that is, A\B = A.\B.
If A is a square matrix, A\B is roughly the same as inv(A)*B, except
it is computed in a different way. If A is an n-by-n matrix and B is a
column vector with n elements, or a matrix with several such columns,
then X = A\B is the solution to the equation AX = B. A warning message
is displayed if A is badly scaled or nearly singular.
If A is an m-by-n matrix with m ~= n and B is a column vector with m
components, or a matrix with several such columns, then X = A\B is the
solution in the least squares sense to the under- or overdetermined
system of equations AX = B. In other words, X minimizes norm(A*X - B),
the length of the vector AX - B. The rank k of A is determined from
the QR decomposition with column pivoting. The computed solution X has
at most k nonzero elements per column. If k < n, this is usually not
the same solution as x = pinv(A)*B, which returns a least squares
solution.
mrdivide(B,A) and the equivalent B/A perform matrix right division
(forward slash). B and A must have the same number of columns.
If A is a square matrix, B/A is roughly the same as B*inv(A). If A is
an n-by-n matrix and B is a row vector with n elements, or a matrix
with several such rows, then X = B/A is the solution to the equation
XA = B computed by Gaussian elimination with partial pivoting. A
warning message is displayed if A is badly scaled or nearly singular.
If B is an m-by-n matrix with m ~= n and A is a column vector with m
components, or a matrix with several such columns, then X = B/A is the
solution in the least squares sense to the under- or overdetermined
system of equations XA = B.