Answer:
a) The set of solutions is y b) the set of solutions is
.
Step-by-step explanation:
a) Let's first find the echelon form of the matrix .
- We add
from row 1 to row 2 and we obtain the matrix
- From the previous matrix, we multiply row 1 by
and the row 2 by
and we obtain the matrix
. This matrix is the echelon form of the initial matrix.
The system has a free variable (x3).
- x2+3x3=0, then x2=-3x3
- 0=x1+
x3=
x1+(-3x3)+
x3=
x1-x3+
x3
then x1=0.
The system has infinite solutions of the form (x1,x2,x3)=(0,-3x3,x3), where x3 is a real number.
b) Let's first find the echelon form of the aumented matrix .
- To row 2 we subtract row 1 and we obtain the matrix
- From the previous matrix, we add to row 3,
of row 2 and we obtain the matrix
.
- From the previous matrix, we multiply row 2 by
and we obtain the matrix
. This matrix is the echelon form of the initial matrix.
The system has a free variable (x4).
- x3-
x4=
, then x3=
x3+
x4=
x2=x4.
- x1-2x2+x3-4x4=1, x1+
+
x4+
The system has infinite solutions of the form (x1,x2,x3,x4)=(-6,x4,
+
x4,x4), where x4 is a real number.