1. Let

Using Matlab (each question below can be answered by one Matlab statement),

1) calculate b^Tb ;

2) calculate (b + jb)^T AA^T b ;

3) calculate b^TA - (A^T b)^T

4) calculate (A^T A)^-1 A;

5) solve the equation A^T Ax = c ;

6) solve the equation c^r = xA ^T A ;

7) find the roots of f (s) ;

8) find the roots of f ² (S) ;

9) find the roots of polynomial (s² + 2)f (s) + 30 ;

10) find the roots of polynomial f²(s) + 5 f (s) + 1.

2. Count the number of positive values in a matrix 0.5-rand(5,6) and find out which element in the matrix has the largest value.

3. Write a program that use Matlab ‘function' to do the following calculations:

(a) the sum of all elements in a matrix;

(b) the mean of all negative elements in the first column of a matrix.

4. Write a program that will read in a text string mixed with numbers and letters from the keyboard and count how many numbers there are in the string.

5. Plot a 3D graphics described by the following equation:

i = 10sin(0.05ft)

where i is the current in ampere, the time t changes from 0 to 20 seconds and the frequency f can change from 10 to 15 Hz.

6. In the circuit below, if the resistor R₄ changes from 0? to 40? in 0.2? steps and Vs changes from 4v to 24V in 0.1V steps, write a program to plot 3D graphics of the voltage on R₂ against both the resistance of R4 and the voltage Vs using Matlab.

7. In the RLC series circuit below, Vm=240, f=50Hz, R=5?, L=4mH and C=1500µF. Write a program to plot 2D graphics of the current i against time t (from 0 to 0.04s with 0.001steps).

8. In the integrating amplifier circuit below, the input Vi=1.5sin(0.5πt), R1= 20?, C=0.02F. Write a program to plot the output Vo and input Vi against time t from 0 to 8s with 0.02s steps.

9. In the RLC network given below, assume that the input e(t) is a unit-step signal, C=0.0025F, R=2? and the inductance L changes from 0H to 0.3H with 0.03H steps. Write a program to display all unit-step responses of the capacitor's voltage Vc for different inductance L in a 2D diagram.