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Byju's Answer
Standard XII
Mathematics
Rank of a Matrix
Examine the c...
Question
Examine the consistency of the system of equations
x
+
3
y
=
5
,
2
x
+
6
y
=
8
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Solution
Given system of equations
x
+
3
y
=
5
2
x
+
6
y
=
8
This can be written as
A
X
=
B
where
A
=
[
1
3
2
6
]
,
X
=
[
x
y
]
,
B
=
[
5
8
]
Here,
|
A
|
=
6
−
6
=
0
Since,
|
A
|
=
0
Hence, the system has either infinitely many solution (consistent) or no solution (inconsistent)
Now, we will find
(
a
d
j
A
)
B
C
11
=
(
−
1
)
1
+
1
6
=
6
C
12
=
(
−
1
)
1
+
2
2
=
−
2
C
21
=
(
−
1
)
2
+
1
3
=
−
3
C
22
=
(
−
1
)
2
+
2
1
=
1
So, the co-factor matrix is
C
=
[
6
−
2
−
3
1
]
⇒
a
d
j
A
=
C
T
=
[
6
−
3
−
2
1
]
Now,
(
a
d
j
A
)
B
=
[
6
−
3
−
2
1
]
[
5
8
]
=
[
30
−
24
−
10
+
8
]
=
[
6
−
2
]
⇒
(
a
d
j
A
)
B
≠
O
Hence, the system of equation is inconsistent.
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Similar questions
Q.
Examine the consistency of the system of equations.
x
+ 3
y
= 5
2
x
+ 6
y
= 8