In SWMI on S : 1 . Write a matrix equation that determines the loop currents from Figure \"l 1. Solve the system for the loop currents. (10 points) In 2. The following statements are either true in all cases. or false in at least one case. If false. construct a specific counterexample to show that the statement is not always true. It a statement is true. give a justification (not just an example): 2:: .1\" a. If v1, ....v.. are in IR\" and v3 = 21:, + v2. then {15. v2.v3.v..} is W 4" linearly dependent. (8 points) Figure i {imp Current from Textbook b. If v1, "\"174 are in IR\" and v3 = 0. then {V1.1'3.V3. v..} is linearly dependent. (8 points) c. If v1. v; are in R' and v; is not a scalar multiple of '01. then {191.19,} is linearly independent. (8 points) d. If :91, v. are in lili' and v; is not a linear combination of 19,, v;. v... then {171.122.123.134} is linearly independent. (3 points) 3. Suppose an economy has four sectors: Agriculture (A). Energy E. Manufacturing (M). and Transportation (T). Sector A sells 10% of its output to E and 25% to M and retains the rest. Sector E sells 30% of its output to A. 30% to M. and 15% to T and retains the rest. Sector M sells 30%: of its output to A. 1 5% to E. and 35% to T and retains the rest. Sector 1' sells 20% of its output to A. 30% to E. and 10% to M and retains the rest. Copyright 2022 Post University. ALL RIGHTS RESERVE 1:. Construct the exchange table for this economy. (I points) In. Find a set of equilibrium prices for the economy. (8 points) 4. Justify and explain the following: I. Show "18' If \"18 OOIUI'HI'II Of 8 are linearly UQDOI'IUOI'IL thl'l 80 are \"'18 columns of AB. is points) b. Suppose CA. = In. Show that the equation Ax = O has only the trivial eoiution. Explain why A cannot have more columns than rows. (8 polnte) c. Suppose AD = 1,... Show that for any b in Iv". the equation Ax solution. Exaplaln why A cannot have more rows than columns. (8 polnts) bhasa :1. Let A be an invertible n x in matrix. and let B be an n. x p matrix. Show that the equation Ax - B has a unique solution A"B. (3 points) 1 2 5. Let A = [1 3]. Construct a 2 x 3 matrix C using only :1. 1. and o as entries. such 1 5 that CA - :3. Compute AC and show that AC .. I3. (10 points)