matrix theory_basic results and techniques_exercise_1.2.10

Find the inverse of each of the following matrices:

1.
\begin{equation}A=
\begin{pmatrix}
1&a&0\\
0&1&0\\
0&b&1\\
\end{pmatrix}
\end{equation}

The determinant of this matrix is 1,so this matrix is invertible.
\begin{equation}
A^{*}=\begin{pmatrix}
A_{11}&A_{21}&A_{31}\\
A_{12}& A_{22}&A_{32}\\
A_{13}&A_{23}&A_{33}\\
\end{pmatrix}=\begin{pmatrix}
1&-a&0\\
0&1&0\\
0&-b&1\\
\end{pmatrix}
\end{equation}
So the inverse matrix is
\begin{equation}
\frac{A^{*}}{|A|}=A^{*}
\end{equation}$\Box$
2.
\begin{equation}B=
\begin{pmatrix}
1&1&0\\
0&1&1\\
0&0&1\\
\end{pmatrix}
\end{equation}
The determinant of this matrix is 1,so this matrix is also invertible.
\begin{equation}
B^{*}=\begin{pmatrix}
1&-1&1\\
0&1&-1\\
0&0&1\\
\end{pmatrix}
\end{equation}So the inverse matrix of $B$ is $B^{*}$.

3.
\begin{equation}
C=\begin{pmatrix}
1&1&0\\
1&1&1\\
0&1&1\\
\end{pmatrix}
\end{equation}
The determinant of $C$ is 0,so this matrix is not invertible.