Let E and M be 3×3 matrices satisfying the system of equations EMT=(EM)T=20 I.  and (E+M)T=17(E−M)T where I denotes identity matrix of order 3. If E2+M2=abI (where a and b are co-prime), then the value of (a+b) is

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### Solution:

${\left(\mathrm{EM}\right)}^{\mathrm{T}}=20\text{\hspace{0.17em}}\mathrm{I}$

Take transpose on both sides

$\begin{array}{l}\mathrm{EM}=20\mathrm{I}\\ \left(\mathrm{E}+\mathrm{M}{\right)}^{\mathrm{T}}=17\left(\mathrm{E}-\mathrm{M}{\right)}^{\mathrm{T}}\\ {\mathrm{E}}^{\mathrm{T}}+{\mathrm{M}}^{\mathrm{T}}=17\left({\mathrm{E}}^{\mathrm{T}}-{\mathrm{M}}^{\mathrm{T}}\right)\\ 16{\mathrm{E}}^{\mathrm{T}}=18{\mathrm{M}}^{\mathrm{T}}\end{array}$

Take transpose on both sides

$16\mathrm{E}=18\text{\hspace{0.17em}}\mathrm{M}$

From Eqns. (1) and (2), we get

$\begin{array}{l}\mathrm{E}=±\frac{3\sqrt{10}}{2}\mathrm{I}; \mathrm{M}=±\frac{4\sqrt{10}}{3}\mathrm{I}\\ {\mathrm{E}}^{2}+{\mathrm{M}}^{2}=\frac{725}{18}\mathrm{I} ⇒\mathrm{a}+\mathrm{b}=743\end{array}$

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