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Question : 9 of 120
Marks:
+1 ,
-0
Solution:
Given system of equations is
3 x 1 + 7 x 2 + x 3 = 2 x 1 + 2 x 2 + x 3 = 3 2 x 1 + 3 x 2 + 4 x 3 = 13 The coefficient matrix
A = | 3 7 1 1 2 1 2 3 4 | ; | A | = | 3 7 1 1 2 1 2 3 4 | : 3 ( 8 − 3 ) − 7 ( 4 − 2 ) + 1 ( 3 − 4 ) = 0 adj ( A ) = [ 5 − 2 − 1 − 25 10 5 5 − 2 − 1 ] T = [ 5 − 25 5 − 2 10 − 2 − 1 5 − 1 ] adj ( A ) B = [ 5 − 25 5 − 2 10 − 2 − 1 5 − 1 ] [ 2 3 13 ] = [ 10 − 75 + 65 − 4 + 30 − 25 − 2 + 15 − 13 ] [ 0 0 0 ]
Hence, it has infinite number of solutions. Alternate method
Augmented matrix
[ A , B ] = [ 3 7 1 ⋮ 2 1 2 1 ⋮ 3 2 3 4 ⋮ 13 ] R 1 ↔ R 2 ∽ [ 1 2 1 ⋮ 3 3 7 1 ⋮ 3 2 3 4 ⋮ 13 ]
Use operations.
R 2 → R 2 − 3 R 1 , R 3 → R 3 − 2 R 1 ∽ [ 1 2 1 ⋮ 3 0 1 − 2 ⋮ − 7 0 − 1 2 ⋮ 7 ] R 3 → R 2 + R 3 ,
∽ [ 1 2 1 ⋮ 3 0 1 − 2 ⋮ − 7 0 0 0 ⋮ 0 ] Here, Rank of
[ A , B ] = Rank of
A So, the system of equation is consistent.
Also. here rank of
A < Number of unknowns i.e.
2 < 3 Hence, the system has infinitely many solutions.
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