Answer
$x=a$ or $x=b$.
Work Step by Step
We know that for a matrix
$
\left[\begin{array}{rrr}
a & b & c \\
d &e & f \\
g &h & i \\
\end{array} \right]
$
the determinant is given as
$D=a(ei-fh)-b(di-fg)+c(dh-eg).$
Thus we have
$D=a((x+b)\cdot 1-x\cdot1)-(b)(x\cdot 1-x\cdot 0)+(x-a)(x\cdot 1-(x+b)\cdot 0)=a(b)-b(x)+(x-a)(x)=ab-bx+x^2-ax=(x-a)(x-b)$.
We know that this is equal to $0$, so $(x-a)(x-b)=0$
Hence,
$x=a$ or $x=b$.
