Problem I need help answering them

1) Perform Gauss-Jordan reduction (using matrices) and present the solution set as a set of vectors. ) One with 4 variables and 3 equations: xty- ztw=3 2x 0 0 - = -w=1 3x+y+= 0 0=0 We apply row operations to the augmented matrix: 0 0 0 ..New Page ......... b) One with 3 variables and 4 equations: xty + 2= = 0 2x- y+ = =1 Ax ty+ 5z = 1 xty - = =0 We apply row operations to the augmented matrix: 0 0 02) Proving a Biconditional Statement 1 Vertical Space (3mm) Prove that, where a, b, c,d are (possibly zero-valued) real parameters, The linear system ax + by : 0 0.x + dy : 0 has a unique solution if and only if ad 7 be y 0 Comment a) To prove a biconditional (p if and only if q) you must prove two conditionals: p a q and q a p . In) One can apply Gauss's method to the linear system but one must consider the cases of certain variables being zero separately. 0) It is easier to deal with an equality than a claim that two expressions are not equal, so consider whether it will be easier to prove the contrapositive of a conditional rather than the conditional itself. Proof: Let a, 1:, 02d 6 R 2 and consider the following linear system: New Page 3) Solve the following two systems. Express each solution set S using vectors. First, identify a particular solution and solve the associated homogeneous system. Express the solution as the sum of this particular solution and a solution of the associated homogeneous system. a) One with two equations: 2x + 4y : 6 4x + 8y : 12 By inspection we see that a particular solution is x : my : The associated homogeneous system can be solved as follows: [DD\\aug0 i [DEHaugO] DD\\aug0 D DEI\\aug0 ................................................. Newpage b) One with lree equations: x + 2y : I] [I : By setting the free variables to zero we see that the particular solution is x : _y : _z : Qw : The associated homogeneous system can be solved as follows: DDEIEHaugO DDEIEHaugO D D DDEIEHaugO 4) Show (according to the definitions in chapter 2) that one the following matrices is nonsingular while the other is singular. Which is which (and why)? (a) ww 2 1 142 WUL ( 6 ) WWW 4 2 Consider the matrix in (a) first and .New Page 5) If these two matrices are row equivalent , show that this is true; if they are not, show that they are not. (Carry this showing out according to the suggested method in the text.) 102 1 1 2 3-1 1 _and_ 0 2 10 5-15 204 Activate Windows Go to Settings to activate Windows. Consider the matrix in the left first and