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

The following is from Chiang's fundamental mathematical economics.

4 0 -1 1. Use the determinant 1 -7 to verify the first four properties of determinants. WN 2. Show that, when all the elements of an nth-order determinant | Aj are multiplied by a number k, the result will be k"| Al. 3. Which properties of determinants enable us to write the following? 9 18 (a) 19 18 27 56 - 0 2 (6) 2 22 = 18 2 3 4. Test whether the following matrices are nonsingular: (a) 19 (C) 13 9 (b) (d) 3 0 W 8 6 5. What can you conclude about the rank of each matrix in Prob. 4? 6. Can any of the given sets of 3-vectors below span the 3-space? Why or why not? (0) [1 2 1] [2 3 1] [3 4 2] (b) [8 1 3] [1 2 8] [-7 1 5] 7. Rewrite the simple national-income model (3.23) in the Ax = d format (with Y as the first variable in the vector x), and then test whether the coefficient matrix A is nonsingular. 8. Comment on the validity of the following statements: (a) "Given any matrix A, we can always derive from it a transpose, and a determinant." (b) "Multiplying each element of an n x n determinant by 2 will double the value of that determinant." (c) "If a square matrix A vanishes, then we can be sure that the equation system Ax = d is nonsingular." 1. Suppose that we expand a fourth-order determinant by its third column and the cofac- tors of the second-column elements. How would you write the resulting sum of prod- ucts in _ notation? What will be the sum of products in ) notation if we expand it by the second row and the cofactors of the fourth-row elements? 2. Find the inverse of each of the following matrices: (0) A = (6) 8=[-3 2] ( c=[3 7] () 0=[6 ;] 3. (a) Drawing on your answers to Prob. 2, formulate a two-step rule for finding the ad- joint of a given 2 x 2 matrix A: In the first step, indicate what should be done to the two diagonal elements of A in order to get the diagonal elements of adj A; in the second step, indicate what should be done to the two off-diagonal elements of A. (Warning: This rule applies only to 2 x 2 matrices.) (b) Add a third step which, in conjunction with the previous two steps, yields the 2 x 2 inverse matrix A-. 4. Find the inverse of each of the following matrices: -2 1 1 (0) E = (C) G = 0 0 N 0 0 0 -1 0 0 (6) F= (d) H = 0 1 0 0