Numerical issues in solving linear systems There was an exam in the class. The course is &quot;High Performance Scientific Computing&quot;. One of the question in the exam is as follows:

Consider the linear system
$$ \begin{bmatrix} a &amp; b \\ b &amp; a \end{bmatrix} \times \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$
with $a,b&gt;0$.
a) If $a$ is very similar to $b$, what is the numerical difficulty in solving this linear system?
b) Suggest a numerically stable formula for computing $z = x + y$ given $a$ and $b$.

This is a Computer Engineering course, however I am not able to answer these questions. What is the keyword to find a solution on the issue?
Thanks in advance.
The $2\times 2$ matrix has determinant close to zero, and so its condition number is very large, causing numerical instability.

The explicit solution of your system is
$$
x=\frac{a}{a^2-b^2}, y=-\frac{b}{a^2-b^2}.
$$
Therefore
$$
x+y = \frac{a-b}{a^2-b^2} = \frac{1}{a+b},
$$
thus avoiding the numerical issues.