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 FigureĀ 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 \(\det I = 1.\)

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Figure3.4.5The determinant \(\det I = 1\) as seen on the left. Otherwise, the determinant \(\det A = -2\) where \(A\) is the matrix whose columns are shown on the right.

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 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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