Please explain how to solve step by step and name any rules or theorems used.

The following is from Chiang's fundamental mathematical economics.

.3 1. Given u' = [5 1 3], v = [3 1 - 1], w = [7 5 8], and x' = [x1 x2 x3], write out the column vectors, u, v, w, and x, and find (0) uv (c) xx (e) u'v (9) u'u (b) uw' (d) v'u (! ) wx (h) x'x 2. Calculate: (a) Ab (b) Alb (c) X'IA (d) x'A Does the insertion of / in (b) affect the result in (a)? Does the deletion of / 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 0 0 a22 . . . . . 0 0 .... Cnn . 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 alto- gether from such a matrix? 6. Since the 3-space is spanned by the three unit vectors defined in (4.7), any other 3-vector should be expressible as a linear combination of el, ez, and es. Show that the following 3-vectors can be so expressed: 25 (a) (b) ( c ) ( d) COON OVA 7. In the three-dimensional Euclidean space, what is the distance between the following points? (0) (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