EXERCISE 1

1. Rewrite the market model (3.1) in the format of (4.1), and show that, if the three variables are arranged in the order Q_d, Q_s, and P, the coefficient matrix will be

How would you write the vector of constants?

2. Rewrite the market model (3.12) in the format of (4.1) with the variables arranged in the following order: Q_d1, Q_s1, Q_d2, Q_s2, P₁, P₂. Write out the coefficient matrix, the variable vector, and the constant vector.

3. Can the market model (3.6) be rewritten in the format of (4.1)? Why?

4. Rewrite the national-income model (3.23) in the format of (4.1), with Y as the first vari¬able. Write out the coefficient matrix and the constant vector.

5. Rewrite the national-income model of Exercise 3.5-1 in the format of (4.1), with the variables in the order Y, T, and C. [Hint: Watch out for the multiplicative expression b(Y - T) in the consumption function.]

EXERCISE 2

1. Givenfind:

(a) A + B (b)C - A (c)3A (d)48 +2C

2. Given

(a) Is AB defined? Calculate AB. Can you calculate BA? Why?
(b) Is BC defined? Calculate BC. Is C8 defined? If so, calculate C8. Is it true that BC = CB?

3. On the basis of the matrices given in Example 9, Is the product BA defined? If so, calculate the product. In this case do we have AB = BA?

4. Find the product matrices in the following (in each case, append beneath every matrix a dimension indicator):

5. In Example 7, if we arrange the quantities and prices as column vectors instead of row vectors, is Q.P defined? Can we express the total purchase cost as Q.P? As Q.P? As Q'.P?

6. Expand the following summation expressions:

(a) Σ_i=2⁵x_i      (d) Σ_i=1ⁿa_ix^i-1

(b) Σ_i=5⁸a_ix_i    (e) Σ_i=0³(x+i)²

(C) Σ_i=1⁴bx

7. Rewrite the following in Σ notation:

(a) x₁ (x₁ - 1) + 2x₂(x₂ - 1) + 3x₃(x₃ - 1)

(b) a₂(x₃ + 2) + a₃(x₄ + 3) + a₄(x₅ + 4)

(c) 1/x + 1/x² + ...... + 1/xⁿ (x ≠ 0)

(d) 1 + x + 1/x² + ..... + 1/xⁿ (x ≠ 0)

8. Show that the following are true:

EXERCISE 3

1. Given u' = [5 1 3], v' = [3 1 -1], w' = [7 5 8], and x' = [x₁ x₂ x₃J, write out the column vectors, u, v, w, and x, and find

(a) uv'   (c) xx'   (e) u'v   (g) u'u
(b) uw'  (d) Vu   (f) w'x   (h) x'x

2. Given

(a) Which of the follow ng are defined: w'x, x'y', xy', y'y, zz', yw', x.y?
(b) Find all the products that are defined.

3. Having sold n items of merchandise at quantities Q₁,...... Qn, and prices P₁,...... P_n, how would you express the total revenue in (a) ∑ notation and (b) vector notation?

4. Given two nonzero vectors w₁ and w₂, the angle θ(0^o ≤ θ ≤ 180^o) they form is related to the scalar product w₁'w₂ (= w₂‘w₁) as follows:

θ is a(n) angle if and only if w^'₁,w₂^' 0

Verify this by computing the scalar product for each of the following pair of vectors (see Figs. 4.2 and 4.3):

5. Given find the following graphically:

(a) 2v           (c) u - v      (e) 2u + 3v
(b) u +v        (d) v -u       (f) 4u - 2v 

6. Since 3-space is spanned by the three unit vectors linear combination of e₁, e₂, and e₃. Show that the following 3-vectors can be so expressed:  

7. In the three-dimensional Euclidean space, what is the distance between the following points?

(a) (3, 2, 8) and (0, -1, 5) (b) (9, 0, 4) and (2, 0, -4)

8. The triangular inequality is written with the weak inequality sign ≤, rather than the strict Inequality sign <. Under what circumstances would the "=" part of the inequal¬ity apply?

9. Express the length of a radius vector v in the Euclidean n-space (i.e., the distance from the origin to point v) by using each of the following:

(a) scalars (b) a scalar product (c) an inner product

EXERCISE 4

1. Given, verify that

(a) (A + B) + C = A +(B +C)
(b) (A + B)- C = A + (B - C)

2. The subtraction of a matrix B may be considered as the addition of the matrix ( -1)B. Does the commutative law of addition permit us to state that A - B = B -- A? If not, how would you correct the statement?

3. Test the associative law of multiplication with the following matrices:

4. Prove that for any two scalars g and k
(a) k(A B) = kA + kB
(b) (g + k)A = gA + k A
(Note: To prove a result, you cannot use specific examples.)

5. For (a) through (d find C = AB.

6. Prove that (A+ B)(C + D). AC + AD+ BC + BD.

7. If the matrix A in Example 5 had all its four elements nonzero, would x'Ax still give a weighted sum of squares? Would the associative law still apply?

8. Name some situations or contexts where the notion of a weighted or unweighted sum of squares may be relevant.

EXERCISE 5

Given

1. Calculate: (a) AI (b) IA (c) Ix (d) x'l
Indicate the dimension of the identity matrix used in each cast

2. Calculate: (a) Ab (b) AIb (c) x'I A (d) x'A
Does the insertion of I in (b) affect the result in (a)? Does the deletion of I in (d) affect the result in (c)?

3. What is the dimension of the null matrix resulting from each of the following?
(a) Premultiply A by a 5 x 2 null matrix.
(b) Postmultiply A by a 3 x 6 null matrix.
(c) Premultiply b by a 2 x 3 null matrix.
(d) Postmultiply x by a 1 x 5 null matrix.

4. Show that the diagonal matrix

can be idempotent only if each diagonal element is either 1 or 0. How many different numerical idempotent diagonal matrices of dimension n x n can be constructed altogether from such a matrix?

EXERCISE 6

1. Given find A', B', and C'.

2. Use the matrices given in Prob. 1 to verify that
(a) (A + B)' = A' + B' (b) (AC)' = C'A'

3. Generalize the result (4.11) to the case of a product of three matrices by proving that, for any conformable matrices A, B, and C, the equation (A B C)' = C'B'A' holds.

4. Given the following four matrices, test whether any one of them is the inverse of another:

5. Generalize the result (4. 4) by proving that, for any conformable nonsingular matrices A, 8, and C, the equation (ABC)^-1 = C^-1B^-1A^-1 holds.

6. Let A = 1 - X(X' X)^-1 X'.

(a) Must A be square? Must (X'X) be square? Must X be square?

(b) Show that matrix A is idempotent. [Note: If X' and X are not square, it is inappro-priate to apply (4.14).