 Solution

For $A+\gamma B$ with $\gamma \ge 0$, the three Gershgorin discs have positive real centres $a+\gamma$, $2c+2\gamma$, $6b+6\gamma$, and radii $b+c$, $a+b+\gamma$, $a+c+\gamma$. Centre minus radius is respectivelyThus none of the discs contains zero. By the Gershgorin circle theorem, zero is not an eigenvalue, so $A+\gamma B$ is nonsingular for every $\gamma \ge 0$. Consequently