Question 1 Answer the following questions based on the matrices ...
Question 1 Answer the following questions based on the matrices given below.
A=
B=
C=
D=
(a) What is the dimension of DA' ? What is the (2,3) element inside DA'? (6 marks)
(b) What is the dimension of C'+D ? What is the (2,1) element inside C'+D? (6 marks)
(c) Define "idempotent matrix". Is matrix AB' idempotent? (8 marks)
Question 2: Answer each of the following. (a) Prove that (AB)' = B'A' given A has as many columns as B has rows (8 marks)
(b) Find the area of the parallelogram shown below using "determinant". (6 marks)
(c) Suppose matrix A is 8x5, matrix B is 8x8, and matrix C is such that C' = B. . Given the above, what is the dimension of matrix W if W = B(AA')C? Is W symmetric?
Question 3: Consider the matrices A and B below
(a) What must the values of w and y inside A be for A to be symmetric? (4 marks)
Consider next the quadratic form z=x'Ax, where A is the matrix given above. (b) How many terms does z have? Explain. (4 marks)
(c) What is the coefficient of the terms x1*x2 and x2*x4 inside z? (6 marks)
Suppose next q = A s and s = B' t, where q, s and t are matrices with one column whose elements are variables and A and B are the matrices given above. (d) How many variables are inside t ? (e) What is the coefficient of variable _______ in the equation that shows the dependence of variable ______ on the ___t___ variables? (4 marks)
Question 4: Consider the following vectors:
(a) Is vector v longer than vector z ? ? (4 marks)
(b) Determine whether vectors v and w are orthogonal? (4 marks)
(c) Draw these three vectors on the same graph. (6 marks)
(d) Depict on your graph the solution to the system of linear equations given below without deriving the solution algebraically and comment on whether the solution values will be negative or positive. (6 marks)
Question 5
where Y, C and T are the variables in the model.
(a) Express the system in matrix form such that the matrix of coefficients is 3x3. (6 marks)
(b) Find the determinant of the matrix of coefficients. (4 marks)
(c) Find the solution values of variables C and T using Cramer's rule. (10 marks)
Question 6 Answers the following questions using the functions given below.
(a) Find the first-order derivative of each function. (8 marks)
(b) Which function above, if any, is both concave and convex? (4 marks)
(c) For x>0, is function f concave or convex? (4 marks)
(c) For x>0, is function g concave or convex? (4 marks)
A=
B=
C=
D=
(a) What is the dimension of DA' ? What is the (2,3) element inside DA'? (6 marks)
(b) What is the dimension of C'+D ? What is the (2,1) element inside C'+D? (6 marks)
(c) Define "idempotent matrix". Is matrix AB' idempotent? (8 marks)
Question 2: Answer each of the following. (a) Prove that (AB)' = B'A' given A has as many columns as B has rows (8 marks)
(b) Find the area of the parallelogram shown below using "determinant". (6 marks)
(c) Suppose matrix A is 8x5, matrix B is 8x8, and matrix C is such that C' = B. . Given the above, what is the dimension of matrix W if W = B(AA')C? Is W symmetric?
Question 3: Consider the matrices A and B below
(a) What must the values of w and y inside A be for A to be symmetric? (4 marks)
Consider next the quadratic form z=x'Ax, where A is the matrix given above. (b) How many terms does z have? Explain. (4 marks)
(c) What is the coefficient of the terms x1*x2 and x2*x4 inside z? (6 marks)
Suppose next q = A s and s = B' t, where q, s and t are matrices with one column whose elements are variables and A and B are the matrices given above. (d) How many variables are inside t ? (e) What is the coefficient of variable _______ in the equation that shows the dependence of variable ______ on the ___t___ variables? (4 marks)
Question 4: Consider the following vectors:
(a) Is vector v longer than vector z ? ? (4 marks)
(b) Determine whether vectors v and w are orthogonal? (4 marks)
(c) Draw these three vectors on the same graph. (6 marks)
(d) Depict on your graph the solution to the system of linear equations given below without deriving the solution algebraically and comment on whether the solution values will be negative or positive. (6 marks)
Question 5
where Y, C and T are the variables in the model.
(a) Express the system in matrix form such that the matrix of coefficients is 3x3. (6 marks)
(b) Find the determinant of the matrix of coefficients. (4 marks)
(c) Find the solution values of variables C and T using Cramer's rule. (10 marks)
Question 6 Answers the following questions using the functions given below.
(a) Find the first-order derivative of each function. (8 marks)
(b) Which function above, if any, is both concave and convex? (4 marks)
(c) For x>0, is function f concave or convex? (4 marks)
(c) For x>0, is function g concave or convex? (4 marks)
