Chapter 7.4, Problem 2P

(a)

To determine

To prove: The equation $\frac{dW}{dt}=\left|\begin{array}{cc}\frac{d{x}_1^{\left(1\right)}}{dt}& \frac{d{x}_1^{\left(2\right)}}{dt}\\ x_2^{\left(1\right)}& x_2^{\left(2\right)}\end{array}\right|+\left|\begin{array}{cc}{x}_1^{\left(1\right)}& x_1^{\left(2\right)}\\ \frac{d{x}_2^{\left(1\right)}}{dt}& \frac{d{x}_2^{\left(2\right)}}{dt}\end{array}\right|$.

(b)

To determine

To prove: The equation $\frac{dW}{dt}=\left(p_{11}+p_{22}\right)W$.

(c)

To determine

The value of $W\left(t\right)$ by solving $\frac{dW}{dt}=\left(p_{11}+p_{22}\right)W$ and obtain a conclusion.

(d)

To determine

To prove: The theorem for any arbitrary values of n.