Let \(\mathcal {H}\) and \(\mathcal {K}\) be complex separable infinite dimensional Hilbert spaces. Given relations \(A\in \mathcal {B}\mathcal {R}(\mathcal {H})\) and \(B\in \mathcal {B}\mathcal {R}(\mathcal {K})\) , we define \(M_{X}:= \begin{bmatrix}{\begin{smallmatrix} A& X\\ 0& B \end{smallmatrix}}\end{bmatrix}\) where \(X\in \mathcal {B}\mathcal {R}(\mathcal {K},\mathcal {H})\) is an unknown relation. In this paper, a necessary and sufficient condition is given for \(M_{X}\) to be an invertible (a left invertible or a right invertible) relation for some \(X\in \mathcal {B}\mathcal {R}(\mathcal {K},\mathcal {H})\) . Moreover, the perturbations \(\begin{array}{l} \bigcap \limits _{X\in \mathcal {B}\mathcal {R}(\mathcal {K},\mathcal {H})} \sigma (M_{X}),\, \bigcap \limits _{X\in \mathcal {B}\mathcal {R}(\mathcal {K},\mathcal {H})} \sigma _{l}(M_{X}),\, \bigcap \limits _{X\in \mathcal {B}\mathcal {R}(\mathcal {K},\mathcal {H})} \sigma _{\delta }(M_{X}) \end{array} \) are also characterized. Finally, the defect sets \(\begin{array}{l} (\sigma _{\star }([A\ X(0)]_{\mathcal {H}})\cup \sigma _{\star }(B)) \setminus \sigma _{\star }(M_{X}) \end{array}\) are actually described, where \(\sigma _{\star }\) is the spectrum and the left (right) spectrum.
