Answer to Question #236809 in Linear Algebra for moe

Question

You are given the following matrix

 $A=\left(\begin{array}{cccc}a & b & c\\ d & e & f \\g & h & i\end{array} \right).$

Which of the following is the determinant of A?

1. $aei+bfg−cdh−ceg−afh−bdi$
2. $aei+bfg+cdh−ceg−afh+bdi$
3. $aei−bfg+cdh−ceg+afh−bdi$
4. $aei+bfg+cdh−ceg−afh−bdi$
5. $−aei−bfg−cdh+ceg+afh+bdi$

Answer 1

There is a square matrix $A=$$\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$. By defenition $det(A)$ 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) $a_{11}a_{22}a_{33}$

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) $a_{12}a_{23}a_{31}$

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) $a_{13}a_{21}a_{32}$

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) $a_{13}a_{22}a_{31}$

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) $a_{11}a_{23}a_{32}$

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) $a_{12}a_{21}a_{33}$

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 $det(A)=a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32} -a_{12}a_{21}a_{33}$

If there is $A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ , using the defenition of determinant for matrix 3x3 and calculete it:

$det(A)=aei+bfg+cdh-ceg-afh-bdi$

It is answer number 4.

Answer: 4. $aei+bfg+cdh-ceg-afh-bdi$