Area Scale Factor and the Determinant

Find the area scale factor of $\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right)$.

• 1det = ($2\times 2$) − ($0\times 0$) = $4$
• 2Area scale factor = |det| = $4$

Answer: $4$ — matching the fact that a linear scale factor of $2$ multiplies area by $2^2=4$

Find the area scale factor of the reflection matrix $\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$.

• 1det = ($1\times -1$) − ($0\times 0$) = $-1$
• 2Area scale factor = |det| = |$-1$| = $1$

Answer: $1$ — a reflection does not change a shape's size, only its orientation. The negative sign in the determinant signals the mirror-image flip, not a size change.