Question

Consider the system of lienar equations: ar +by = c dr + ey=f (a) Write the above system in the form Arc = b (b) Show that the system has a unique solution if and only if A has an inverse. Find the unique solution. (c) The corresponding homogenous equation is given by Az = 0 i.e. set the right hand side of the equation to zero). Show that the homogenous equation has a non-trivial solution if and only if A does not have an inverse. In this case, the solutions are either the entire plane or a line through the origin. (d) Determine h and k shuch that the solution set of system a) is empty, b) contains a unique solution, and c) contains infinitely many solutions: x1 + 3x2 = k 4.01 +h22 = 8

Answer 1

1 axtby = c } can be written an a dx tey = f f (ab 19 = 10) TA b) Let A has Invesse then AX = B ginn AT (AX) = A' ZATA) X= AB (A Concatriya -- IxAB (Pyvese property) X A1B (properly of Identi a ATB is the unique solation anversely A = B han unique solution ->> Rank (A) 2 det A 0 » A ý non singular = A han Prverse. " A" Non singular ( A has Inverse - AX=B has unique solution c) Consider AX=0 L A has Invesse & A CAX) = Ao f x=0 2) Ax=b has zero solution (Tourial solution - Ax=b han non fornal Solutim G A has e no inverse =) Rank (A) <2 ) Rank (A=|| & Solution : axot by C & solution in entire plane cor a line through Orisin if e=0.

d (131K 14ul & 13 | K Non-128-4k) (i) It has unique solution if det A to I det lo sell to > h 12 40 h $12 7) It has On finitely many solution TAIFO and f(A) = CARB) = 1 h-12 = and fraK=0 - n = 12 and R=2. ill) No solution of SEA) & F1A1B) » h = 12 and K #2