###### Example 3.4.4

Consider the determinant of the identity matrix

\begin{equation*}
I =
\left[\begin{array}{rr} 1\amp 0 \\ 0 \amp 1 \\
\end{array}\right]
=
\left[\begin{array}{rr} \evec_1 \amp \evec_2 \\
\end{array}\right]\text{.}
\end{equation*}

As seen on the left of FigureĀ 3.4.5, the vectors \(\vvec_1 = \evec_1\) and \(\vvec_2=\evec_2\) form a positively oriented pair. Since the parallelogram they form is a \(1\times1\) square, we have \(\det I = 1.\)

Now we will consider the matrix

\begin{equation*}
A =
\left[\begin{array}{rr} -2\amp 0 \\ 0 \amp 1 \\
\end{array}\right]
=
\left[\begin{array}{rr} \vvec_1 \amp \vvec_2 \\
\end{array}\right]\text{.}
\end{equation*}

As seen on the right of FigureĀ 3.4.5, the vectors \(\vvec_1\) and \(\vvec_2\) form a negatively oriented pair. The parallelogram they define is a \(2\times1\) rectangle so we have \(\det A = -2\text{.}\)