12. Work with 2 × 2 matrices as transformations of the plane and interpret the absolute value of the determinant in terms of area.
Students should appreciate the ease with which they can find the determinant of a 2 × 2 matrix. Seriously. It's super easy. The determinant of matrix:
is ad – bc. That's it.
Let's say we have a parallelogram with its vertices at coordinates (0, 0), (a, b), (a + c, b + d), and (c, d). Then the area of that parallelogram is the absolute value of the determinant of the matrix:
Deep breaths. It's easier than it looks.
If we have a parallelogram with vertices at O (0, 0), P (2, 4), Q (3, 9), and R (1, 5), we can make the matrix:
Its determinant is 2 × 5 – 1 × 4 = 10 – 4 = 6. And the absolute value of 6 is also 6. (Duh.) So the area of parallelogram OPQR is 6 units2.
Students should also know that you aren't making this up. If they don't believe you, make them check by calculating the area of that same parallelogram the old-fashioned way. Make them use an abacus, while you're at it. That'll teach them to doubt you.