prove-that-exists-X-M-2-3-R-Y-M-3-2-R-such-that-X-Y-1-1-1-1-Y-X-2-6-6-3-9-9-3-9-9-

Question Number 219562 by hardmath last updated on 28/Apr/25

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{exists}\:\:\:\mathrm{X}\:\in\:\mathrm{M}_{\mathrm{2},\mathrm{3}} \:\left(\mathbb{R}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Y}\:\in\:\mathrm{M}_{\mathrm{3},\mathrm{2}} \:\left(\mathbb{R}\right) \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\:\mathrm{X}\centerdot\mathrm{Y}\:=\:\begin{pmatrix}{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{1}}\end{pmatrix}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Y}\centerdot\mathrm{X}\:=\:\begin{pmatrix}{\mathrm{2}}&{\mathrm{6}}&{\mathrm{6}}\\{\mathrm{3}}&{\mathrm{9}}&{\mathrm{9}}\\{-\mathrm{3}}&{-\mathrm{9}}&{-\mathrm{9}}\end{pmatrix}\: \\ $$

Answered by vnm last updated on 30/Apr/25

$${X}=\begin{bmatrix}{{a}}&{\mathrm{3}{a}}&{\mathrm{3}{a}}\\{{a}}&{\mathrm{3}{a}}&{\mathrm{3}{a}}\end{bmatrix} \\ $$$${Y}=\begin{bmatrix}{\mathrm{1}/{a}}&{\mathrm{1}/{a}}\\{\mathrm{3}/\left(\mathrm{2}{a}\right)}&{\mathrm{3}/\left(\mathrm{2}{a}\right)}\\{−\mathrm{3}/\left(\mathrm{2}{a}\right)}&{−\mathrm{3}/\left(\mathrm{2}{a}\right)}\end{bmatrix} \\ $$

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