##### Example3.4.4

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 Figure 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 Figure 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 $$\det A = -2\text{.}$$

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