Given-that-det-a-b-c-d-e-f-g-h-i-n-find-det-d-2a-e-2b-f-2c-2a-2b-2c-4g-4h-4i-

Question Number 210080 by Spillover last updated on 30/Jul/24

$${Given}\:{that}\:\:{det}\:\begin{bmatrix}{{a}}&{{b}}&{{c}}\\{{d}}&{{e}}&{{f}}\\{{g}}&{{h}}&{{i}}\end{bmatrix}={n} \\ $$$$ \\ $$$${find}\:{det}\begin{bmatrix}{{d}+\mathrm{2}{a}}&{{e}+\mathrm{2}{b}}&{{f}+\mathrm{2}{c}}\\{\mathrm{2}{a}}&{\mathrm{2}{b}}&{\mathrm{2}{c}}\\{\mathrm{4}{g}}&{\mathrm{4}{h}}&{\mathrm{4}{i}}\end{bmatrix} \\ $$$$ \\ $$

Commented by Frix last updated on 30/Jul/24

$$\mathrm{Doesn}'\mathrm{t}\:\mathrm{give}\:\mathrm{any}\:“\mathrm{nice}''\:\mathrm{result}. \\ $$

Commented by Spillover last updated on 30/Jul/24

$${what}\:{result}\:{did}\:{you}\:{get}? \\ $$

Commented by Frix last updated on 30/Jul/24

$$\mathrm{Sorry}\:\mathrm{I}\:\mathrm{had}\:\mathrm{a}\:\mathrm{typo}\:\mathrm{in}\:\mathrm{my}\:\mathrm{calculation}. \\ $$$$\mathrm{The}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{det}\:=−\mathrm{8}{n} \\ $$

Commented by Spillover last updated on 30/Jul/24

$${correct} \\ $$

Answered by Spillover last updated on 30/Jul/24

$${det}\:\begin{bmatrix}{{a}}&{{b}}&{{c}}\\{{d}}&{{e}}&{{f}}\\{{g}}&{{h}}&{{i}}\end{bmatrix}={n} \\ $$$$ \\ $$$${det}\:\begin{bmatrix}{{a}}&{{b}}&{{c}}\\{{d}}&{{e}}&{{f}}\\{\mathrm{4}{g}}&{\mathrm{4}{h}}&{\mathrm{4}{i}}\end{bmatrix}=\mathrm{4}{n} \\ $$$$ \\ $$$${det}\:\begin{bmatrix}{\mathrm{2}{a}}&{\mathrm{2}{b}}&{\mathrm{2}{c}}\\{{d}}&{{e}}&{{f}}\\{\mathrm{4}{g}}&{\mathrm{4}{h}}&{\mathrm{4}{i}}\end{bmatrix}=\mathrm{8}{n} \\ $$$$ \\ $$$${det}\:\begin{bmatrix}{\mathrm{2}{a}}&{\mathrm{2}{b}}&{\mathrm{2}{c}}\\{{d}+\mathrm{2}{a}}&{{e}+\mathrm{2}{b}}&{{f}+\mathrm{2}{c}}\\{\mathrm{4}{g}}&{\mathrm{4}{h}}&{\mathrm{4}{i}}\end{bmatrix}=\mathrm{8}{n} \\ $$$$ \\ $$$${det}\:\begin{bmatrix}{\mathrm{2}{a}+{d}}&{\mathrm{2}{b}+{e}}&{\mathrm{2}{c}+{f}}\\{\mathrm{2}{a}}&{\mathrm{2}{b}}&{\mathrm{2}{c}}\\{\mathrm{4}{g}}&{\mathrm{4}{h}}&{\mathrm{4}{i}}\end{bmatrix}=−\mathrm{8}{n} \\ $$$$ \\ $$$$ \\ $$

Commented by Frix last updated on 30/Jul/24

$$\mathrm{Yes}. \\ $$

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