# Common Core Standards: Math

#### The Standards

# High School: Number and Quantity

### Vector and Matrix Quantities HSN-VM.C.12

**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 units^{2}.

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.