Question 10

10

(i)

We have:

\begin{align*}\left(\begin{matrix}2 & 3 \\ 3 &2\end{matrix}\left|\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right.\right) &\xrightarrow {r_2 \leftrightarrow r_1} \left(\begin{matrix}3 & 2 \\ 2 & 3\end{matrix}\left|\begin{matrix}0 & 1 \\ 1 & 0\end{matrix}\right.\right) \\ &\xrightarrow {r_1 \to 2r_1, r_2 \to 3r_2} \left(\begin{matrix}6 & 4 \\ 6 & 9\end{matrix}\left|\begin{matrix}0 & 2 \\ 3 & 0\end{matrix}\right.\right) \\ & \xrightarrow {r_2 \to r_2 - r_1} \left(\begin{matrix}6 & 4 \\ 0 & 5\end{matrix}\left|\begin{matrix}0 & 2 \\ 3 & -2\end{matrix}\right.\right) \\ & \xrightarrow{r_2 \to \frac 1 5 r_2} \left(\begin{matrix}6 & 4 \\ 0 & 1\end{matrix}\left|\begin{matrix}0 & 2 \\ \frac 3 5 & -\frac 2 5\end{matrix}\right.\right) \\ &\xrightarrow{r_2 \to 4 r_2} \left(\begin{matrix}6 & 4 \\ 0 & 4\end{matrix}\left|\begin{matrix}0 & 2 \\ \frac {12} 5 & -\frac 8 5\end{matrix}\right.\right) \\ &\xrightarrow {r_1 \to r_1 - r_2} \left(\begin{matrix}6 & 0 \\ 0 & 4\end{matrix}\left|\begin{matrix}-\frac {12} 5 & \frac {18} 5 \\ \frac {12} 5 & -\frac 8 5\end{matrix}\right.\right) \\ &\xrightarrow {r_1 \to \frac 1 6 r_1, r_2 \to \frac 1 4 r_2} \left(\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\left|\begin{matrix}-\frac 2 5 & \frac 3 5 \\ \frac 3 5 & -\frac 2 5\end{matrix}\right.\right)\end{align*}

So:

\begin{pmatrix}2 & 3 \\ 3 & 2\end{pmatrix}^{-1} = \begin{pmatrix}-\frac 2 5 & \frac 3 5 \\ \frac 3 5 & -\frac 2 5\end{pmatrix}

(ii)

\begin{align*}\left(\begin{matrix}1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1\end{matrix} \left|\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right.\right) &\xrightarrow {r_3 \to r_3 - r_2} \left(\begin{matrix}1 & 1 & 0 \\ 1 & 0 & 1 \\ -1 & 1 & 0\end{matrix} \left|\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1\end{matrix}\right.\right) \\ & \xrightarrow {r_3 \to r_1 + r_3} \left(\begin{matrix}1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 2 & 0\end{matrix} \left|\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & -1 & 1\end{matrix}\right.\right) \\ & \xrightarrow {r_3 \to \frac 1 2 r_3} \left(\begin{matrix}1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{matrix} \left|\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ \frac 1 2 & -\frac 1 2 & \frac 1 2\end{matrix}\right.\right) \\ & \xrightarrow {r_1 \to r_1 - r_3} \left(\begin{matrix}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{matrix} \left|\begin{matrix}\frac 1 2 & \frac 1 2 & -\frac 1 2 \\ 0 & 1 & 0 \\ \frac 1 2 & -\frac 1 2 & \frac 1 2\end{matrix}\right.\right) \\ & \xrightarrow {r_2 \to r_2 - r_1} \left(\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{matrix} \left|\begin{matrix}\frac 1 2 & \frac 1 2 & -\frac 1 2 \\ -\frac 1 2 & \frac 1 2 & \frac 1 2 \\ \frac 1 2 & -\frac 1 2 & \frac 1 2\end{matrix}\right.\right) \\ &\xrightarrow {\text{swapping rows}} \left(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \left|\begin{matrix}\frac 1 2 & \frac 1 2 & -\frac 1 2 \\ \frac 1 2 & -\frac 1 2 & \frac 1 2 \\ -\frac 1 2 & \frac 1 2 & \frac 1 2\end{matrix}\right.\right)\end{align*}

So we have:

\begin{pmatrix}1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1\end{pmatrix}^{-1} = \begin{pmatrix}\frac 1 2 & \frac 1 2 & -\frac 1 2 \\ \frac 1 2 & -\frac 1 2 & \frac 1 2 \\ -\frac 1 2 & \frac 1 2 & \frac 1 2\end{pmatrix}

(iii)

\begin{align*}\left(\begin{matrix}1 & - 1 & 0 & 0 \\ 1 & 0 & - 1 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 1 & 1\end{matrix}\ \left|\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{matrix}\right.\right) & \xrightarrow {r_4 \to r_1 + r_2 + r_3} \left(\begin{matrix}1 & - 1 & 0 & 0 \\ 1 & 0 & - 1 & 0 \\ 1 & 0 & 0 & -1 \\ 3 & 0 & 0 & 0\end{matrix}\ \left|\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1\end{matrix}\right.\right) \\ & \xrightarrow {r_4 \to \frac 1 3 r_4} \left(\begin{matrix}1 & - 1 & 0 & 0 \\ 1 & 0 & - 1 & 0 \\ 1 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0\end{matrix}\ \left|\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \frac 1 3 & \frac 1 3 & \frac 1 3 & \frac 1 3\end{matrix}\right.\right) \\ &\xrightarrow {r_1 \leftrightarrow r_4} \left(\begin{matrix}1 & 0 & 0 & 0 \\ 1 & - 1 & 0 & 0 \\ 1 & 0 & - 1 & 0 \\ 1 & 0 & 0 & -1\end{matrix}\ \left|\begin{matrix}\frac 1 3 & \frac 1 3 & \frac 1 3 & \frac 1 3 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{matrix}\right.\right) \\ &\xrightarrow {r_2 \to r_2 - r_1} \left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 1 & 0 & - 1 & 0 \\ 1 & 0 & 0 & -1\end{matrix}\ \left|\begin{matrix}\frac 1 3 & \frac 1 3 & \frac 1 3 & \frac 1 3 \\ \frac 2 3 & -\frac 1 3 & -\frac 1 3 & -\frac 1 3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{matrix}\right.\right) \\ & \xrightarrow {r_3 \to r_3 - r_1} \left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 1 & 0 & 0 & -1\end{matrix}\ \left|\begin{matrix}\frac 1 3 & \frac 1 3 & \frac 1 3 & \frac 1 3 \\ \frac 2 3 & -\frac 1 3 & -\frac 1 3 & -\frac 1 3 \\ -\frac 1 3 & \frac 2 3 & -\frac 1 3 & -\frac 1 3 \\ 0 & 0 & 1 & 0\end{matrix}\right.\right) \\ & \xrightarrow {r_4 \to r_4 - r_1} \left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & -1\end{matrix}\ \left|\begin{matrix}\frac 1 3 & \frac 1 3 & \frac 1 3 & \frac 1 3 \\ \frac 2 3 & -\frac 1 3 & -\frac 1 3 & -\frac 1 3 \\ -\frac 1 3 & \frac 2 3 & -\frac 1 3 & -\frac 1 3 \\ -\frac 1 3 & -\frac 1 3 & \frac 2 3 & -\frac 1 3\end{matrix}\right.\right) \\ & \to \left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{matrix}\ \left|\begin{matrix}\frac 1 3 & \frac 1 3 & \frac 1 3 & \frac 1 3 \\ -\frac 2 3 & \frac 1 3 & \frac 1 3 & \frac 1 3 \\ \frac 1 3 & -\frac 2 3 & \frac 1 3 & \frac 1 3 \\ \frac 1 3 & \frac 1 3 & -\frac 2 3 & \frac 1 3\end{matrix}\right.\right)\end{align*}

So that:

\begin{pmatrix}1 & -1 & 0 & 0 \\ 1 & 0 & -1 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 1 & 1\end{pmatrix}^{-1} = \begin{pmatrix}\frac 1 3 & \frac 1 3 & \frac 1 3 & \frac 1 3 \\ -\frac 2 3 & \frac 1 3 & \frac 1 3 & \frac 1 3 \\ \frac 1 3 & -\frac 2 3 & \frac 1 3 & \frac 1 3 \\ \frac 1 3 & \frac 1 3 & -\frac 2 3 & \frac 1 3\end{pmatrix}

Augmented matrices take ages in latex lol.