Solution
= Solution
To solve $A\mathbf x=\mathbf b$, subtract the first row from the second, and twice the first from the third. The resulting equations give $y=b_2-b_1$, $y+z=b_3-2b_1$, and hence $z=b_3-b_1-b_2$. Substituting in the original first equation gives $x=7b_1-3b_3$. Thus the <inverse matrix> is
$$
\boxed{A^{-1}=\begin{pmatrix}7&0&-3\\-1&1&0\\-1&-1&1\end{pmatrix}.}
$$
The <determinant> is one, and multiplying in either order gives the <identity matrix>.