1.2.1

(1, 0, 0)

1.2.2

(1, 2, 3)

1.2.3

(1, 0, -1)

1.2.4

(-1/2, 1/2, 0, 1)

1.2.5

1.2.6

Every row operation is reversible. In particular the "inverse" of any row operation is again a row operation of the same type.

1.2.7

?/2, ?, 0

1.2.8

The third equation in the triangularized form is 0 x ₃ = 1, which is impossible to solve.

1.2.9

The third equation in the triangularized form is 0 x ₃ = 0, and all numbers are solutions. This means that you can start the back substitution with any value whatsoever and consequently produce infinitely many solutions for the system.

1.2.10

? = -3, ? = 11/2, and ? = -3/2

1.2.11

1. If x _i = the number initially in chamber # i, then

   
   

   and the solution is x ₁ = 10 , x ₂ = 20, x ₃ = 30, and x ₄ = 40.

2. 16, 22, 22, 40

1.2.12

To interchange rows i and j, perform the following sequence of Type II and Type III operations.

1.2.13

(a) This has the effect of interchanging the order of the unknowns- x _j and x _k are permuted. (b) The solution to the new system is the same as the solution to the old system except...