Prove that the determinant of this matrix is non-zero.

Question

Prove that the determinant of this matrix is non-zero for every possible combination of + and - .$$\left[\begin{array}{cc} \pm 1 & \pm 3 & \pm 4 \\ \pm 3 & \pm 2 & \pm 5 \\ \pm 4 & \pm 5 & \pm 3 \end{array}\right]$$

Answer 1

Considering all possible combinations you have that $$D=(\pm6\pm25)\pm3(\pm9\pm20)\pm4(\pm15\pm8)=\{\pm31,\pm19\}\pm\{\pm87,\pm33\}\pm\{\pm92,\pm28\}$$ where the $,$ in the $\{,\}$ should be interpreted as "or".

Now since the summand of the last $\{\}$ is even you should try to cancel it out with a combination of numbers of the two brackets. But after a not very tedious inspection you can see that no combination of them yields neither $\pm92$ (this is almost immediate) nor $\pm28$ so that the even term of the last term cannot be canceled out. Hence $D\neq0$.