answer:A perfectly competitive firm produces at a level where marginal cost (MC) equals the market price, as price is also equal to marginal revenue (MR) in this market structure. This occurs because the firm faces a perfectly elastic demand curve, leading to a situation where maximizing profit or minimizing loss requires MC to be equal to MR. On the other hand, a monopolist produces where price (P) exceeds marginal cost. This is because the monopolist has market power and can charge a higher price than a competitive firm due to its control over the supply. Monopolists face a downward-sloping demand curve, causing their marginal revenue to be less than the price, and they maximize profit where MR equals MC, which is at a point where P is above MC.

answer:begin{array}{l} text{Replace the second row of the matrix with the vector (4, 0, -1):} left( begin{array}{ccc} 2 & 9 & -2 4 & 0 & -1 2 & -6 & 9 end{array} right) text{Calculate the determinant of the resulting matrix:} left| begin{array}{ccc} 2 & 9 & -2 4 & 0 & -1 2 & -6 & 9 end{array} right| = -2(0-6) - 9(36+2) - (-2)(-24-0) = boxed{-322} end{array}

answer:Considering that ( h ), ( phi ), and ( n ) are independent, we can analyze the distribution of the real part ( y_r ). A complex Gaussian random variable ( x sim mathcal{CN}(0, sigma^2) ) has circular symmetry, meaning for any ( theta in [-pi, pi) ), ( e^{itheta}x ) has the same distribution as ( x ): [ e^{itheta}x stackrel{d}{=}x ] Thus, we can rewrite ( y ) and maintain the same distribution: [ y = e^{iphi} h + n stackrel{d}{=} h + n ] Since ( h ) and ( n ) are both complex Gaussian and independent, their sum ( y ) is also complex Gaussian with a combined variance: [ y sim mathcal{CN}(0, sigma_h^2 + sigma_n^2) ] The real and imaginary parts of a complex Gaussian random variable are independent and normally distributed with equal variances. Therefore, the real part ( y_r ) is: [ y_r sim mathcal{N}(0, frac{sigma_h^2 + sigma_n^2}{2}) ] Hence, the pdf of ( y_r ) is ( mathcal{N}(0, frac{sigma_h^2 + sigma_n^2}{2}) ).

answer:The development of various dog breeds can be attributed to intentional crossbreeding, where dogs from distinct regions were bred together. This process, over time, resulted in the genetic combination and diversification that led to the many unique breeds we see today.