|1/3 4| |1/6 3/2| find the value of determinant
Answers
Answer:
the determinat is 0
Step-by-step explanation:
Since column 3 of the given matrix consists of numbers which are all constant multiples of each other, there is a theorem which states that the determinant of this matrix is
0
.
However, I will prove it from calculation just to verify and show how you would calculate such determinants should you not be aware of the theorems of linear matrix algebra.
Use co-factor expansion along any row or column of your choice.
This involves adding the products of the entries in each row with their co-factors
(
−
1
)
i
+
j
, for an entry in row I and column j, multiplied by the minor of the entry, formed by evaluating the resulting determinant when you delete row I and column j.
In this case, since the given matrix is
3
×
3
in dimension, it will result in three
2
×
2
determinants which we can find from definition
∣
∣
∣
a
b
c
d
∣
∣
∣
=
a
d
−
b
c
I will take co-factor expansion along row 1 as an example to find the determinant of the given matrix as :
Δ
=
1
(
−
1
)
1
+
1
∣
∣
∣
5
6
8
9
∣
∣
∣
+
2
(
−
1
)
1
+
2
∣
∣
∣
4
6
7
9
∣
∣
∣
+
3
(
−
1
)
1
+
3
∣
∣
∣
4
5
7
8
∣
∣
∣
=
1
(
45
−
48
)
+
(
−
2
)
(
36
−
42
)
+
3
(
32
−
35
)
=
−
3
+
12
−
9
=
0