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

The following is from Chiang's fundamental mathematical economics.

1. Use Cramer's rule to solve the following equation systems: (a) 3x1 - 2x2 = 6 (c) 8x1 - 7x2 = 9 2x1 + x2 = 11 X1 + x2 = 3 (b) -x + 3x2 = -3 (d) 5x1 + 9x2 = 14 4x1 - X2 = 12 7x1 - 3X2 = 4 2. For each of the equation systems in Prob. 1, find the inverse of the coefficient matrix, and get the solution by the formula x* = Ad. 3. Use Cramer's rule to solve the following equation systems: (a) 8x1 - x2 = 16 (c) 4x + 3y -2z=1 2x2 + 5x3 = 5 x + 2y =6 2x1 + 3x3 = 7 3x + 1=4 (b) -x1 + 3x2 + 2x3 = 24 ( d) -x + y+1=0 + *3 = 6 x - y+z=b 5x2 - X3 = 8 X + y - Z = C 4. Show that Cramer's rule can be derived alternatively by the following procedure. Mul- tiply both sides of the first equation in the system Ax = d by the cofactor |C1/1, and then multiply both sides of the second equation by the cofactor |C2/l, etc. Add all the newly obtained equations. Then assign the values 1, 2, ..., n to the index j, succes- sively, to get the solution values x;, x2, ..., x; as shown in (5.17). 3. Let the IS equation be A Y = 1-b 1-b where 1 - b is the marginal propensity to save, g is the investment sensitivity to inter- est rates, and A is an aggregate of exogenous variables. Let the LM equation be Y = k where k and / are income and interest sensitivity of money demand, respectively, and Mo is real money balances. If b = 0.7, g = 100, A = 252, k = 0.25, / = 200, and Mo = 176, then (a) Write the IS-LM system in matrix form. (b) Solve for Y and i by matrix inversion. 1. On the basis of the model in (5.24), if the final demands are d; = 30, dy = 15, and di = 10 (all in billions of dollars), what are the solution output levels for the three in- dustries? (Round off answers to two decimal places.) 2. Using the information in (5.23), calculate the total amount of primary input required to produce the solution output levels of Prob. 1. 3. In a two-industry economy, it is known that industry I uses 10 cents of its own product and 60 cents of commodity II to produce a dollar's worth of commodity I; industry !I uses none of its own product but uses 50 cents of commodity I in producing a dollar's worth of commodity ll; and the open sector demands $1,000 billion of commodity I and $2,000 billion of commodity II. () Write out the input matrix, the Leontief matrix, and the specific input-output matrix equation for this economy. (b) Check whether the data in this problem satisfy the Hawkins-Simon condition. (c) Find the solution output levels by Cramer's rule. 4. Given the input matrix and the final-demand vector 0.05 0.25 0.34 1800 A = 0.33 0.10 0.12 200 0.19 0.38 0 900_ (a) Explain the economic meaning of the elements 0.33, 0, and 200. (b) Explain the economic meaning (if any) of the third-column sum. (c) Explain the economic meaning (if any) of the third-row sum. (d) Write out the specific input-output matrix equation for this model. (e) Check whether the data given in this problem satisfy the Hawkins-Simon condition. 5. (a) Given a 4 x 4 matrix 8 = [b/], write out all the principal minors. (b) Write out all the leading principal minors. 6. Show that, by itself (without other restrictions on matrix 8), the Hawkins-Simon condi- tion already guarantees the existence of a unique solution vector x", though not nec- essarily nonnegative