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.