Show that each of the following systems of linear equations is consistent and also find their
Solution:
x + y + z = 6
x + 2y + 3z = 14
x + 4y + 7z = 30
This can be written as:
|A| = 1(2) – 1(4) + 1(2)
= 2 – 4 + 2
|A| = 0
So, A is singular. Thus, the given system is either inconsistent or it is consistent with infinitely many solution according to as:
(Adj A)x B≠0 or (Adj A)x B = 0
Cofactors of A are:
C11 = (– 1)1 + 1 14 – 12 = 2
C21 = (– 1)2 + 1 7 – 4 = – 3
C31 = (– 1)3 + 1 3 – 2 = 1
C12 = (– 1)1 + 2 7 – 3 = – 4
C22 = (– 1)2 + 1 7 – 1 = 6
C32 = (– 1)3 + 1 3 – 1 = 2
C13 = (– 1)1 + 2 4 – 2 = 2
C23 = (– 1)2 + 1 4 – 1 = – 3
C33 = (– 1)3 + 1 2 – 1 = 1
adj A =
=
Adj A x B =
Now, AX = B has infinite many solution
Let z = k
Then, x + y = 6 – k
X + 2y = 14 – 3k
This can be written as:
|A| = 1
Adj A =
Now, X = A – 1B =
=
=
There values of x,y,z satisfy the third equation
Hence, x = k – 2, y = 8 – 2k, z = k