In this paper, an upper bound on the condition number of the matrix \((I+W)\) is derived, where \(W=X^{-1}Y\) and X & Y are tridiagonal matrices. Such matrices are ubiquitous in numerical solution of partial differential equations. Specifically, the condition number of \((I+W)\) is crucial while solving the parabolic PDEs using compact schemes. To show the application of this bound in numerical PDE, the compact scheme is considered to discretize the convection-diffusion equations. An upper bound on \(\left\Vert W \right\Vert _2\) is obtained using Gerschgorin circle theorem in terms of entries of the matrices X and Y. This is utilized to derive the condition number of the matrix \((I+W)\) which is shown to be of order \(\mathcal {O}\left( \frac{\delta v}{\delta z^2}\right) \) , where \(\delta v\) and \(\delta z\) are time and space step sizes respectively. The numerical illustrations have been added to supplement the theoretical findings. Note that the upper bounds obtained are general, and applicable for other problems as well.