You are given the following matrix
Which of the following is the determinant of A?
There is a square matrix . By defenition is a sum all of products of 3 elements. The elements should be taken one at a time from each row and column. A sign of product depending on the permutation.
Take a products, in each of them indexes by rows have no inversion:
1)
Indexes by row: 1,2,3: no inversion -- 0
Indexes by column: 1,2,3: no inversions -- 0.
0 + 0 = 0, it's even number, sign is plus.
2)
Indexes by row: 1,2,3: no inversions -- 0
Indexes by column: 2,3,1
2 and 1 are inversed, 3 and 1 are inversed -- 2 inversions.
0 + 2 = 2, it's even number, sign is plus.
3)
Indexes by row: 1,2,3: no inversions -- 0
Indexes by column: 3,1,2
3 and 1 are inversed, 3 and 2 are inversed -- 2 inversions.
0 + 2 = 2, it's even number, sign is plus.
4)
Indexes by row: 1,2,3: no inversions -- 0
Indexes by column: 3,2,1
3 and 2 are inversed, 3 and 1 are inversed, 2 and 1 are inversed -- 3 inversions.
0 + 3 = 3, it's odd number, sign is minus.
5)
Indexes by row: 1,2,3: no inversions -- 0
Indexes by column: 1,3,2
3 and 2 are inversed -- 1 inversion.
0 + 1 = 1, it's odd number, sign is minus.
6)
Indexes by row: 1,2,3: no inversions -- 0
Indexes by column: 2,1,3
2 and 1 are inversed -- 1 inversion.
0 + 1 = 1, it's odd number, sign is minus.
So, the
If there is , using the defenition of determinant for matrix 3x3 and calculete it:
It is answer number 4.
Answer: 4.
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