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Properties of Determinants

Prove that th...

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Prove that though covariance is independent of the choice of origin, it depends upon the scale. If u=ax+b, v=cy+d, show that cov(u,v)=a.c. cov(x,y)

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C o v (X, Y) = 6

U = \frac{X - 1}{2}, V = \frac{Y - 2}{3}

c o v (U, V) =

u_{i} = \frac{x_{i} - a}{h}

v_{i} = \frac{y_{i} - b}{k}

then relation between

c o v (u, v)

c o v (x, y)

Covariance of X and Y is given by the formula, Cov(X,Y) = ___

u (x)

v (x)

are two independent solutions of the differential equation

\frac{d^{2} y}{d x^{2}} + b \frac{d y}{d x} + c y = 0

, then additional solution(s) of the given differential equation is(are):

u = a x^{2} + 2 b x y + c y^{2}, u^{'} = a^{'} x^{2} + 2 b^{'} x y + c^{'} y^{2}

, then prove that

∣ ∣ ∣ ∣ \begin{matrix} y^{2} & - x y & x^{2} a & b & c a^{'} & b^{'} & c^{'} \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ \begin{matrix} a x + b y & b x + c y a^{'} x + b^{'} y & b^{'} x + c^{'} y \end{matrix} ∣ ∣ ∣ = - \frac{1}{y} ∣ ∣ ∣ \begin{matrix} u & u^{'} a x + b y & a^{'} x + b^{'} y \end{matrix} ∣ ∣ ∣

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Properties of Determinants

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