1. ## [SOLVED] matrix determinant

if we have the following matrix
2 1 0 0
3 2 0 0
1 1 3 4
2 -1 2 3 and we want to calculate the determinant .shouldn't in any correct way of calculation obtain the same value of determinant?
if we change the 4th column by making c4----- 3c4 -4c3 we will have determinant =3
if we change the 3rd column by making c3--------- 4c3- 3c2 we will have determinant= 4
if we change in lines we will have determinant = 1

2. Originally Posted by euclid2
if we have the following matrix
2 1 0 0
3 2 0 0
1 1 3 4
2 -1 2 3 and we want to calculate the determinant .shouldn't in any correct way of calculation obtain the same value of determinant?
if we change the 4th column by making c4----- 3c4 -4c3 we will have determinant =3
if we change the 3rd column by making c3--------- 4c3- 3c2 we will have determinant= 4
if we change in lines we will have determinant = 1

If B results from A by interchanging two rows or columns, then det(B)=-det(A)
If B results from A by multiplying one row or column with the number c, then det(B)=c*det(A)
If B results from A by adding a multiple of one row to another row, or a multiple of one column to another column, then det(B)=det(A)

sorry about the first post, apparently i can't copy and paste formulae

3. Originally Posted by euclid2
if we have the following matrix
2 1 0 0
3 2 0 0
1 1 3 4
2 -1 2 3 and we want to calculate the determinant .shouldn't in any correct way of calculation obtain the same value of determinant?
if we change the 4th column by making c4----- 3c4 -4c3 we will have determinant =3
if we change the 3rd column by making c3--------- 4c3- 3c2 we will have determinant= 4
if we change in lines we will have determinant = 1
Consider this:

(1) we add the product of column 3 by -4 to the column 4

2 1 0 0
3 2 0 0
1 1 3 -8
2 -1 2 -5

or

(2) we add the product of column 2 by -3 to the column 3

2 1 -3 0
3 2 -6 0
1 1 0 4
2 -1 5 3

If we multiply a column/row by a factor the determinant is multiplied by the same factor.

In your two examples the determinant is multiplied by

(1) 3

(2) 4

Another way to compute the determinant is to transform the original matrix into a upper or lower triangular matrix. Starting with

Matrix A

2 1 0 0
3 2 0 0
1 1 3 4
2 -1 2 3

det A=1

and applying the pivot technique we get the following equivalent Lower Triangular Matrix

Matrix LTM

1/2 0 0 0
3 2 0 0
-5/3 7/3 1/3 0
2 -1 2 3

det LTM = product of the main diagonal entries = (1/2)×2×(1/3)×3 = 1

4. Originally Posted by TheMasterMind
Consider this:

(1) we add the product of column 3 by -4 to the column 4

2 1 0 0
3 2 0 0
1 1 3 -8
2 -1 2 -5

or

(2) we add the product of column 2 by -3 to the column 3

2 1 -3 0
3 2 -6 0
1 1 0 4
2 -1 5 3

If we multiply a column/row by a factor the determinant is multiplied by the same factor.

In your two examples the determinant is multiplied by

(1) 3

(2) 4

Another way to compute the determinant is to transform the original matrix into a upper or lower triangular matrix. Starting with

Matrix A

2 1 0 0
3 2 0 0
1 1 3 4
2 -1 2 3

det A=1

and applying the pivot technique we get the following equivalent Lower Triangular Matrix

Matrix LTM

1/2 0 0 0
3 2 0 0
-5/3 7/3 1/3 0
2 -1 2 3

det LTM = product of the main diagonal entries = (1/2)×2×(1/3)×3 = 1
So the determinant is 1?....Thanks

5. Originally Posted by euclid2
So the determinant is 1?....Thanks