need help on all three questions

41. [5 pts.) Use the conversion table in Lecture Notes If}, and the key matrix ti 2:1 1 A = 13 Hi 1|] 2|] 1? 15 to encrypt the massage: \"STUDY IJNEAR ALGEBEA." Note that the blank spaces and the period must also be encoded. 5-. [5 pta.) For a clmed Leontief economic model with 2 sectors, we seek to solve the equilibrium equation Ax = x, where A is the 2 x 2 consumption matrix, and x E IR2 is the output {or production} vector. Show that ifs = [l 1], then sA = s. {Recall that the columns of A sum to 1.} EL. {It} phi.) For a closed Leontiei' economic model with n sectors, we seek to solve the equilibrium equation Ax = x, where A is the u x u consumption matrix, and x E IR" is the output {or production] vector. Show that the equilibrium equation always has a nonzero solution. [Hint Recall that the columns of A sum to 1. Show that the [colunm} vector 5 E R" whose entries are all equal to 1, is a solution to Arx = x. Next, argue by way of contradiction that if Ax = x has only the zero solution, then the matrix [I A)?\" is invertible. Why is this a contradiction