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# Determinant Help! Watch

1. if the matrix A is defined as

a d g
b e h
c f i

and det(A) = 1

Then,

Matrix B defined as

(a+d) (d+g) (g+a)
(b+e) (e+h) (h+b)
(c+f) ( f+i) ( i+c)

det(B) = ?

I know the basic rules regarding determinants such as row interchange = det sign change, adding multiples of other rows have no effect and that constants change the det by the value of the constant.

I'm inclined to say the det is still 1 in B since col 1 is just col1+colb, col 2 is col2+col3, col 3 is col3+col1

Is this right?
2. (Original post by Zilch)
if the matrix A is defined as

a d g
b e h
c f i

and det(A) = 1

Then,

Matrix B defined as

a+d d+g g+a
b+e e+h h+b
c+f f+i i+c

det(B) = ?

I know the basic rules regarding determinants such as row interchange = det sign change, adding multiples of other rows have no effect and that constants change the det by the value of the constant.

I'm inclined to say the det is still 1 in B since col 1 is just col1+colb, col 2 is col2+col3, col 3 is col3+col1

Is this right?
Yep, it's one as adding columns doesn't affect the determinant
3. (Original post by cpdavis)
Yep, it's one as adding columns doesn't affect the determinant
Isn't this true if atleast one of the original rows still remain though? If you change all of them, I think the det does change?
4. (Original post by Zilch)
Isn't this true if atleast one of the original rows still remain though? If you change all of them, I think the det does change?
Nope, adding and taking away doesn't affect the determinant. If you wanted to, you can look at it from the Elementary matrices and you'll see that this is true This is why row/column reduction doesn't change the determinant

If you want to see it works, you can try it with the identity matrix as an example
5. (Original post by cpdavis)
Nope, adding and taking away doesn't affect the determinant. If you wanted to, you can look at it from the Elementary matrices and you'll see that this is true This is why row/column reduction doesn't change the determinant

If you want to see it works, you can try it with the identity matrix as an example
Consider the matrix

1 1 1
0 0 1
1 0 1

This has det = 1

After I do Row 2 + row 1 = Row2 and Row 3+Row 2 = Row 3, the det is still the same.

However, after I do Row 1+row3= Row1, the det changes to 2. Note that I am using the old row 3, before my previous row operations.

I would have thought that it would still stay the same?
6. (Original post by Zilch)
Consider the matrix

1 1 1
0 0 1
1 0 1

This has det = 1

After I do Row 2 + row 1 = Row2 and Row 3+Row 2 = Row 3, the det is still the same.

However, after I do Row 1+row3= Row1, the det changes to 2. Note that I am using the old row 3, before my previous row operations.

I would have thought that it would still stay the same?
I got 1 and not 2
7. (Original post by cpdavis)
I got 1 and not 2
My bad, I was using the old row 2 again for my second row operation. thanks!

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Updated: April 19, 2013
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