If you know how to multiply 2x2 matrices, and know about
complex numbers, then you'll enjoy this connection. Any
complex number (a+bi) can be represented by a real 2x2
matrix in the following way!
Let the 2x2 matrix
[ a b ]
[ b a ]
correspond to (a+bi). Addition of complex numbers then
corresponds to addition of the corresponding 2x2 matrices.
So does multiplication! Observe if you take this product:
[ a b ] [ c d ]
[ b a ] [ d c ]
you get
[ (acbd) (ad+bc) ]
[ (ad+bc) (acbd) ]
which is precisely what you would get if you multiplied
(a+bi) and (c+di) and then converted to a 2x2 matrix!
Presentation Suggestions:
Let students do the multiplication, or maybe have done
it already for homework before you present this fun fact.
As a follow up Fun Fact, note that taking determinants of these matrices
produce
Products of Sums of Two Squares.
The Math Behind the Fact:
The reason this works is because complex multiplication
can be viewed as a linear transformation on the 2dimensional
plane. In linear algebra, you learn that every linear transformation
can be represented as matrix multiplication by a suitable matrix.
How to Cite this Page:
Su, Francis E., et al. "Really Complex Matrices."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
