Art of transpose matrix

Algebra Level 5

Let E E and M M be 3 × 3 3 \times 3 matrices satisfy the system of equations below and I I denotes the identity matrix:

{ E M T = ( E M ) T = 20 I ( E + M ) T = 17 ( E − M ) T \large \begin{cases} EM^{T} = (EM)^{T} = 20 I \\ (E+M)^{T} = 17(E-M)^{T} \end{cases}

If E 2 + M 2 E^2+M^2 is of the form a b I \dfrac{a}{b}I , where a a and b b are co-prime, find a + b a+b .


The answer is 743.

Tommy Li by Tommy Li , 22, Hong Kong

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1 solution

Tommy Li
Oct 7, 2017

E M T = M T E T = 20 I ⇒ { E = E T = 20 ( M − 1 ) T M = M T = 20 ( E − 1 ) T ( E + M ) T = 17 ( E − M ) T E T + M T = 17 ( E T − M T ) ⇒ { E + 20 E − 1 = 17 ( E − 20 E − 1 ) 20 M − 1 + M = 17 ( 20 M − 1 − M ) ⇒ { E = ± 3 10 2 I M = ± 4 10 3 I E 2 + M 2 = ( ± 3 10 2 I ) 2 + ( ± 4 10 3 I ) 2 = 725 18 I ⇒ a + b = 725 + 18 = 743 EM^T=M^TE^T= 20I \\ \Rightarrow\begin{cases} E=E^T=20(M^{-1})^T \\ M=M^T=20(E^{-1})^T \end{cases} \\ \\ (E+M)^T=17(E-M)^T \\ E^T+M^T=17(E^T-M^T) \\ \Rightarrow \begin{cases} E+20E^{-1} = 17(E-20E^{-1}) \\ 20M^{-1}+M = 17(20M^{-1}-M) \end{cases} \\ \Rightarrow \begin{cases} E = \pm \frac{3\sqrt{10}}{2}I \\ M = \pm \frac{4\sqrt{10}}{3}I \end{cases} \\ E^2+M^2 = \left(\pm \dfrac{3\sqrt{10}}{2}I\right)^2 + \left(\pm \dfrac{4\sqrt{10}}{3}I\right)^2 = \dfrac{725}{18}I \\ \Rightarrow a+b =725+18 =743

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