Young’s modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type

1a. 

Show from first principles, i.e., by using the definition of linear independence,
that if μ = x + iy, y ̸= 0 is an eigenvalue of a real matrix
A with associated eigenvector v = u + iw, then the two real solutions
Y(t) = eat(u cos bt − wsin bt)
and
Z(t) = eat(u sin bt + wcos bt)
are linearly independent solutions of ˙X = AX.

1b. 

Use (a) to solve the system (see image)
 

A car travels at 108 kph from A to B for t seconds, applies the brakes for 4 seconds between B and C to give the car a constant deceleration and a speed of 72 kph at C, and finally travels at this speed from C to D, also for t seconds. If the total horizontal distance from A to D is 3100 m, determine the distance between A and B.

${\displaystyle {\left(x^4{y}^5\right)}^{\frac{1}{4}}{\left(x^8{y}^5\right)}^{\frac{1}{5}}=x^{\frac{j}{5}}{y}^{\frac{k}{4}}}$
In the equation above, j and k are constants. If the equation is true for all positive real values of x and y, what is the value of $j-k$?
A)3
B)4
C)5
D)6