1. State the problem: Compute the determinant of the coefficient matrix for the system $x+y+z=6$, $2x+y+z=9$, $x+2y+z=7$. 2. Identify the coefficient matrix $A$: $$A=\begin{pmatrix}1&1&1\\2&1&1\\1&2&1\end{pmatrix}$$ 3. Use the 3$\times$3 determinant formula: $$\det(A)=a(ei-fh)-b(di-fg)+c(dh-eg)$$ where $a=1$, $b=1$, $c=1$, $d=2$, $e=1$, $f=1$, $g=1$, $h=2$, $i=1$. 4. Compute the needed parts: - $ei-fh=1\cdot1-1\cdot2=1-2=-1$. - $di-fg=2\cdot1-1\cdot1=2-1=1$. - $dh-eg=2\cdot2-1\cdot1=4-1=3$. 5. Substitute into the formula: $$\det(A)=1\cdot(-1)-1\cdot(1)+1\cdot(3)$$ 6. Simplify: $$\det(A)=-1-1+3$$ $$\det(A)=1$$ 7. Final answer: $\det(A)=1$.