First slide

Transpose of matrix

Question

$Let E and M be 3 \times 3 matrices satisfying the system of equations {EM}^{T} = (EM)^{T} = 20 I . and (E + M)^{T} = 17 (E - M)^{T}$
$where I denotes identity matrix of order 3 .$
$If E^{2} + M^{2} = \frac{a}{b} I (where a and b are co-prime), then the value of (a + b) is$

Moderate

Solution

${(EM)}^{T} = 20 I$
Take transpose on both sides
$\begin{array}{l} EM = 20 I \\ (E + M)^{T} = 17 (E - M)^{T} \\ E^{T} + M^{T} = 17 (E^{T} - M^{T}) \\ 16 E^{T} = 18 M^{T} \end{array}$
Take transpose on both sides
$16 E = 18 M$
From Eqns. (1) and (2), we get
$\begin{array}{lr} E = \pm \frac{3 \sqrt{10}}{2} I; M = \pm \frac{4 \sqrt{10}}{3} I \\ E^{2} + M^{2} = \frac{725}{18} I \Rightarrow a + b = 743 \end{array}$

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