Matrice inversabila de ordin 2 cu o conditie

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Cezar Lupu
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Matrice inversabila de ordin 2 cu o conditie

Post by Cezar Lupu »

Fie \( A\in M_{2}(\mathbb{R}) \) o matrice pentru care este satisfacuta inegalitatea:

det\( ^{2}(A-I_{2}) \)+det\( ^{2}(A+I_{2})<2. \)

Sa se arate ca matricea \( A \) este inversabila.
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
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Marius Mainea
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Post by Marius Mainea »

Aplicam CBS

\( 2>\det^2(A-I_2)+\det^2(A+I_2)\geq \frac{1}{2}(\det(A-I_2)+\det(A+I_2)^2=2(1+\det A)^2 \)

De aici \( \det A\neq0 \)
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