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Byju's Answer
Standard XII
Mathematics
Feasible Solution
A company pro...
Question
A company produces three products every day. Their production on a certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using matrix method.
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Solution
Let
x
,
y
and
z
be
the
production
level
of
the
first
,
second
and
third
product
,
respectively
.
According to the question,
x
+
y
+
z
=
45
.
.
.
(
1
)
-
x
+
z
=
8
.
.
.
(
2
)
x
+
z
=
2
y
Since
the
production
of
first
and
third
product
is
twice
the
production
of
second
product
x
-
2
y
+
z
=
0
.
.
.
(
3
)
The given system of equation can be written in matrix form as follows:
1
1
1
-
1
0
1
1
-
2
1
x
y
z
=
45
8
0
A
X
=
B
A
=
1
1
1
-
1
0
1
1
-
2
1
X
=
x
y
z
B
=
45
8
0
Now,
A
=1
-
0
+
2
-
1
-
1
-
1
+
1
2
-
0
=
2
+
2
+
2
=
6
Let C
i
j
be the cofactors of the elements a
i
j
in A=
a
i
j
. Then,
C
11
=
-
1
1
+
1
0
1
-
2
1
=
2
,
C
12
=
-
1
1
+
2
-
1
1
1
1
=
2
,
C
13
=
-
1
1
+
3
-
1
0
1
-
2
=
2
C
21
=
-
1
2
+
1
1
1
-
2
1
=
-
3
,
C
22
=
-
1
2
+
2
1
1
1
1
=
0
,
C
23
=
-
1
2
+
3
1
1
1
-
2
=
3
C
31
=
-
1
3
+
1
1
1
0
1
=
1
,
C
32
=
-
1
3
+
2
1
1
-
1
1
=
-
2
,
C
33
=
-
1
3
+
3
1
1
-
1
0
=
1
adj
A
=
2
2
2
-
3
0
3
1
-
2
1
T
=
2
-
3
1
2
0
-
2
2
3
1
⇒
A
-
1
=
1
A
adj
A
=
1
6
2
-
3
1
2
0
-
2
2
3
1
X
=
A
-
1
B
⇒
X
=
1
6
2
-
3
1
2
0
-
2
2
3
1
45
8
0
⇒
X
=
1
6
90
-
24
+
0
90
+
0
+
0
90
+
24
+
0
⇒
x
y
z
=
1
6
66
90
114
∴
x
=
11
,
y
=
15
and
z
=
19
Thus
,
the
production
level
of
first
,
second
and
third
product
is
11
,
15
and
19
,
respectively
.
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