Limit of determinant/norm

Question

I need to prove that $$\lim_{H\to0} \frac{|\det(H)|}{|H|}=0$$ where H is a 2x2 Matrix. I tried using the formulas: $$\lim_{H\to0} \frac{ad-bc}{\sqrt{a^2+b^2+c^2+d^2}}=0$$ but I am not sure if it is the right method. Can someone help me out?

Answer 1

Just try to square the whole expression. The degree in the numerator will be higher than in the denominator.

Answer 2

Note that each of $|a|,|b|,|c|,|d|$ is bounded above by $\|H\|.$ Thus the expression is no more than

$$\frac{2\|H\|^2}{\|H\|} = 2\|H\| \to 0.$$

Answer 3

Note that $$|ad-bc| \leq \sqrt{a^2+b^2+c^2+d^2}. \sqrt{d^2+c^2} $$ Thus

$$\frac{|ad-bc|}{\sqrt{a^2+b^2+c^2+d^2}} \leq \sqrt{d^2+c^2}$$

your limit is equivalent to a,b,c,d go to 0, in particular d,c go to 0, thus your limit is equals to 0.