If you know how to multiply 2×2 matrices, and know about\u00a0complex numbers, then you’ll enjoy this connection. Any complex number (a+bi) can be represented by a real 2×2 matrix in the following way! Let the 2×2 matrix<\/p>\n\n\n\n
[ \u00a0a b ]\u00a0
[ -b a ]<\/p>\n\n\n\n
correspond to (a+bi). Addition of complex numbers then corresponds to addition of the corresponding 2×2 matrices. So does multiplication! Observe if you take this product:<\/p>\n\n\n\n
[ \u00a0a b ] [ c d ]\u00a0
[ -b a ] [ -d c ]<\/p>\n\n\n\n
you get<\/p>\n\n\n\n
[ \u00a0(ac-bd) (ad+bc) ]\u00a0
[ -(ad+bc) (ac-bd) ]<\/p>\n\n\n\n
which is precisely what you would get if you multiplied (a+bi) and (c+di) and then converted to a 2×2 matrix!<\/p>\n\n\n\n
Presentation\u00a0Suggestions:<\/strong> The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong> How to Cite this Page:<\/strong> Fun Fact suggested by<\/strong>: If you know how to multiply 2×2 matrices, and know about\u00a0complex numbers, then you’ll enjoy this connection. Any complex number...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[57,2,73,101,16],"class_list":["post-501","page","type-page","status-publish","hentry","tag-advanced","tag-algebra","tag-complex-numbers","tag-linear-algebra","tag-matrix"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/501","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=501"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/501\/revisions"}],"predecessor-version":[{"id":1596,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/501\/revisions\/1596"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=501"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=501"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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\u00a0Products of Sums of Two Squares.<\/p>\n\n\n\n
The reason this works is because complex multiplication can be viewed as a linear transformation on the 2-dimensional plane. In\u00a0linear algebra, you learn that every linear transformation can be represented as matrix multiplication by a suitable matrix.<\/p>\n\n\n\n
Su, Francis E., et al. “Really Complex Matrices.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Francis Su<\/p>\n","protected":false},"excerpt":{"rendered":"