Page 56 - A Course in Linear Algebra with Applications

40 Chapter Two: Systems of Linear Equations

Exercises 2.1

In the first three problems find the general solution or else
t
show hat the linear system is inconsistent.

x\ + 2x 2 — 3x 3 + x 4 = 7
-xi + x 2 - x 3 + X4 = 4
2.
+ x 2 — £3 — X4 = 0
+ x 3 - x 4 = - 1
+ 2x2 + %3 — 3^4 = 2

3.
xx + x 2 + 2x 3 = 4
Xi - X 2 — £3 = - 1
2xi — 4x2 — 5x3 = 1

Solve the following homogeneous linear systems
xi + x 2 + x 3 + x 4 = 0
(a) { 2xi + 2x 2 + £3 + £4 = 0
xi + x 2 - x 3 + x 4 = 0
2xi — X2 + 3x3 = 0
(b) { 4xi + 2x 2 + 2 x 3 = 0
-2xi + 5x 2 — 4x3 = 0
5. For which values of t does the following homogeneous linear
system have non-trivial solutions?

12xi - x 2 + X3 = 0
tXi + X3 = 0
X 2 + tx 3 = 0