Rank of a matrix plus a number

Question

Suppose to have a symmetric matrix $A$ with rank equal to $a$. Now let $$ A_1 = A + \frac{x}{2} $$

such that $x$ is a real number different from 0.

Is it always true that rank of $A_1 = a+1$? Can you prove that? In which case this will be true?

Answer 1

This is wrong.

Take any invertible symmetric $n\times n$ matrix, hence of maximal rank $n$; since $A+\frac x2$ has the same size, it cannot have a rank larger than that of $A$.

Moreover, if all entries of $A$ were equal to $-\frac x2$, then rk$(A)$ would be $1$ and rk$(A+\frac x2)$ would be $0$. So you see that the rank can decrease as well.