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From the Fun Fact files, here is a Fun Fact at the Advanced level:

# Really Complex Matrices

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

[  (ac-bd) (ad+bc) ]
[ -(ad+bc) (ac-bd) ]

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 2-dimensional 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>.

Keywords:    complex numbers, linear algebra
Subjects:    algebra
Level:    Advanced
Fun Fact suggested by:   Francis Su
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