Let F be an \(r\times k\)-matrix of bounded holomorphic functions in the unit disc such that the nonzero eigenvalues of \(FF^ *\) are bounded from below by some \(\delta >0\). We prove that there is a bounded holomorphic solution to \(Fu=g\) if g is an r-column of bounded holomorphic functions and the equation is solvable pointwise. If \(r=1\), this is Carleson's corona theorem.