(7.3)
REVISION OF MATRICES
GROUP I
Consider the matrices A and B:

A = 
2 3 0 1 

B = 
1 1 0 3 


1 1 4 1 


1 3 2 5 


0 4 2 2 


2 1 6 0 


1 0 3 3 


2 2 1 0 
1. Using a spreadsheet, calculate: A + B, A * B, Det(A), Det(B), A^{1} e B^{1}
2. Show that (A.B)^{1} = B^{1}.A^{1}
3. Show that (A.B)^{T} = B^{T}.A^{T}
GROUP II
Let the Matrices:
M_{(4,4)} = (1/4) 
1 1 1 1 

O_{(4,4)} = 
0 0 0 0 

I_{(4,4)} = 
1 0 0 0 

1 1 1 1 


0 0 0 0 


0 1 0 0 

1 1 1 1 


0 0 0 0 


0 0 1 0 

1 1 1 1 


0 0 0 0 


0 0 0 1 
1. Verify that the matrix null 0 is idempotent.
2. Verify that the matrix identity I is idempotent.
3. Verify that the matrix M is idempotent.
4. What are the traces of M and I?
5. Calculate the ranks, r, of M and of I.
6. What is the value of the determinant of M and I?
GROUP III
1. Verify that the product Mx, where x is the vector given by x^{T} = (3 4 8 1), is a vector with all elements equal to the arithmetic mean, , of the 4 elements of vector x.
2. Verify that (IM)x is the vector of the deviation.
3. Verify that the sum of the squares of x_{i,} Σ(x_{i}^{2}) can be written as: x^{T}. x
4. Verify that the sum of the squares of the deviations, , can also be written in a matricial form, as: x^{T} (IM)x
GROUP IV
1. Consider the vector 
x = 2 + θ 
where θ is an unknown parameter. 

3 θ 

a) Write the derivative of the vector x
b) Calculate x^{T} x
c) Calculate
d) Show that
2. Consider the vector x = 
2 + 4θ_{1}  5θ_{2} 
where θ_{1} and θ_{2} are two unknown constants. 

1+ θ_{1} + θ_{2 }θ_{1}+4θ_{2} 

a) Write the derivative matrix (take θ_{1 }and θ_{2} as variables)
b) Calculate x^{T} x
c) Transpose
d) Show that the transposed matrix
GROUP V
Consider the following system of 2 equations with 2 unknowns;
5 = 2 A + 3 B
4 = A  2 B
1. Show that the equation system can be written in matrix form as,
Y_{(2,1)} = X_{(2,2)} θ_{(2,1)}
where Y is the vector of the independent terms (5 e 4) of the system,
θ is the vector of the unknowns A and B
and X is the matrix of the coefficients of the unknowns
2. Verify that the solution of the system can be given as θ = (X^{T}X)^{1}X^{T}Y
3. Show that X is a square, non singular matrix, and then that the solution of the system can be θ = X^{1}Y
ESTIMATION OF THE PARAMETERS OF THE YOSHIMOTO AND CLARKE MODEL (1993)
4. Estimate the parameters k, q and r, of the Fox integrated model (IFOX) and of Yoshimoto & Clarke (1993) using the following data:
Year 
Y 
CPUE 
1983 
538 
235 
1984 
638 
131 
1985 
431 
63 
1986 
99 
22 
1987 
37 
8 
1988 
62 
21 
1989 
437 
77 
1990 
146 
28 
1991 
126 
26 
1992 
53 
25 
1993 
91 
41 
1994 
232 
66 
which represent the total annual catches (in tons) and the respective catches by fishing effort unit (kg/fishing day of the fleet PESCRUL) of the stock of Deepwater rose shrimp, Parapenaeus longirostris of the Algarve during the period 1983 to 1994 (Mattos Silva, 1995).
Comment on the obtained results comparing them with those presented in Section 8.20.