Find a determinant of a matrix $M_{(n)x(n)}$ depending on the $x,y,n in mathbb R$
$begingroup$
I had a matrix which I I transformed to zero as many matrix elements as possible and in this time I have: $$y^{n} cdot det begin{vmatrix}
frac{x}{y}+1 & 1 & ... & & & 1\
frac{-x}{y} & frac{x}{y} & & & & 0 \
. & 0 & frac{x}{y} & 0 & ... & . \
. & . & ... & & & . \
. & . & & & & . \
frac{-x}{y} & 0 & ... & & &frac{x}{y} \
end{vmatrix}
$$ Unfortunately I don't know what can I do to find a sollution at this moment. Can you help me with some tips?
linear-algebra
asked Jan 18 at 1:40
MP3129MP3129
871211
$endgroup$
add a comment |
$begingroup$
I had a matrix which I I transformed to zero as many matrix elements as possible and in this time I have: $$y^{n} cdot det begin{vmatrix}
frac{x}{y}+1 & 1 & ... & & & 1\
frac{-x}{y} & frac{x}{y} & & & & 0 \
. & 0 & frac{x}{y} & 0 & ... & . \
. & . & ... & & & . \
. & . & & & & . \
frac{-x}{y} & 0 & ... & & &frac{x}{y} \
end{vmatrix}
$$ Unfortunately I don't know what can I do to find a sollution at this moment. Can you help me with some tips?
linear-algebra
asked Jan 18 at 1:40
MP3129MP3129
871211
$endgroup$
add a comment |
$begingroup$
I had a matrix which I I transformed to zero as many matrix elements as possible and in this time I have: $$y^{n} cdot det begin{vmatrix}
frac{x}{y}+1 & 1 & ... & & & 1\
frac{-x}{y} & frac{x}{y} & & & & 0 \
. & 0 & frac{x}{y} & 0 & ... & . \
. & . & ... & & & . \
. & . & & & & . \
frac{-x}{y} & 0 & ... & & &frac{x}{y} \
end{vmatrix}
$$ Unfortunately I don't know what can I do to find a sollution at this moment. Can you help me with some tips?
linear-algebra
asked Jan 18 at 1:40
MP3129MP3129
871211
$endgroup$
I had a matrix which I I transformed to zero as many matrix elements as possible and in this time I have: $$y^{n} cdot det begin{vmatrix}
frac{x}{y}+1 & 1 & ... & & & 1\
frac{-x}{y} & frac{x}{y} & & & & 0 \
. & 0 & frac{x}{y} & 0 & ... & . \
. & . & ... & & & . \
. & . & & & & . \
frac{-x}{y} & 0 & ... & & &frac{x}{y} \
end{vmatrix}
$$ Unfortunately I don't know what can I do to find a sollution at this moment. Can you help me with some tips?
linear-algebra
linear-algebra
asked Jan 18 at 1:40
MP3129MP3129
871211
asked Jan 18 at 1:40
MP3129MP3129
871211
asked Jan 18 at 1:40
MP3129MP3129
871211
asked Jan 18 at 1:40
MP3129MP3129
871211
asked Jan 18 at 1:40
MP3129MP3129
871211
871211
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
(Assumed that your calculations so far are correct,) you're almost there.
Just add the second, third, ... $n$th columns to the first column, so that your matrix will become an upper triangle matrix and its determinant is the product of the diagonal elements.
answered Jan 18 at 1:54
BerciBerci
62k23776
$endgroup$
$begingroup$
In my previous transformations, I used row operations. Can I now transform columns?
$endgroup$
– MP3129
Jan 18 at 2:12
2
$begingroup$
Row transformations retain the set of roots of system corresponding to the matrix, column transformations don't. However iff your aim is just calculating the determinant you can mix row and column transformations.
$endgroup$
– P Vanchinathan
Jan 18 at 2:54
$begingroup$
Using $det(A^T) =det A$, we see that the column operations change the determinant the same way as the row operations.
$endgroup$
– Berci
Jan 18 at 11:02
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
oldest
votes
$begingroup$
(Assumed that your calculations so far are correct,) you're almost there.
Just add the second, third, ... $n$th columns to the first column, so that your matrix will become an upper triangle matrix and its determinant is the product of the diagonal elements.
answered Jan 18 at 1:54
BerciBerci
62k23776
$endgroup$
$begingroup$
In my previous transformations, I used row operations. Can I now transform columns?
$endgroup$
– MP3129
Jan 18 at 2:12
2
$begingroup$
Row transformations retain the set of roots of system corresponding to the matrix, column transformations don't. However iff your aim is just calculating the determinant you can mix row and column transformations.
$endgroup$
– P Vanchinathan
Jan 18 at 2:54
$begingroup$
Using $det(A^T) =det A$, we see that the column operations change the determinant the same way as the row operations.
$endgroup$
– Berci
Jan 18 at 11:02
add a comment |
$begingroup$
(Assumed that your calculations so far are correct,) you're almost there.
Just add the second, third, ... $n$th columns to the first column, so that your matrix will become an upper triangle matrix and its determinant is the product of the diagonal elements.
answered Jan 18 at 1:54
BerciBerci
62k23776
$endgroup$
$begingroup$
In my previous transformations, I used row operations. Can I now transform columns?
$endgroup$
– MP3129
Jan 18 at 2:12
2
$begingroup$
Row transformations retain the set of roots of system corresponding to the matrix, column transformations don't. However iff your aim is just calculating the determinant you can mix row and column transformations.
$endgroup$
– P Vanchinathan
Jan 18 at 2:54
$begingroup$
Using $det(A^T) =det A$, we see that the column operations change the determinant the same way as the row operations.
$endgroup$
– Berci
Jan 18 at 11:02
add a comment |
$begingroup$
(Assumed that your calculations so far are correct,) you're almost there.
Just add the second, third, ... $n$th columns to the first column, so that your matrix will become an upper triangle matrix and its determinant is the product of the diagonal elements.
answered Jan 18 at 1:54
BerciBerci
62k23776
$endgroup$
(Assumed that your calculations so far are correct,) you're almost there.
Just add the second, third, ... $n$th columns to the first column, so that your matrix will become an upper triangle matrix and its determinant is the product of the diagonal elements.
answered Jan 18 at 1:54
BerciBerci
62k23776
answered Jan 18 at 1:54
BerciBerci
62k23776
answered Jan 18 at 1:54
BerciBerci
62k23776
answered Jan 18 at 1:54
BerciBerci
62k23776
62k23776
$begingroup$
In my previous transformations, I used row operations. Can I now transform columns?
$endgroup$
– MP3129
Jan 18 at 2:12
2
$begingroup$
Row transformations retain the set of roots of system corresponding to the matrix, column transformations don't. However iff your aim is just calculating the determinant you can mix row and column transformations.
$endgroup$
– P Vanchinathan
Jan 18 at 2:54
$begingroup$
Using $det(A^T) =det A$, we see that the column operations change the determinant the same way as the row operations.
$endgroup$
– Berci
Jan 18 at 11:02
add a comment |
$begingroup$
In my previous transformations, I used row operations. Can I now transform columns?
$endgroup$
– MP3129
Jan 18 at 2:12
2
$begingroup$
Row transformations retain the set of roots of system corresponding to the matrix, column transformations don't. However iff your aim is just calculating the determinant you can mix row and column transformations.
$endgroup$
– P Vanchinathan
Jan 18 at 2:54
$begingroup$
Using $det(A^T) =det A$, we see that the column operations change the determinant the same way as the row operations.
$endgroup$
– Berci
Jan 18 at 11:02
$begingroup$
In my previous transformations, I used row operations. Can I now transform columns?
$endgroup$
– MP3129
Jan 18 at 2:12
$begingroup$
In my previous transformations, I used row operations. Can I now transform columns?
$endgroup$
– MP3129
Jan 18 at 2:12
2
2
$begingroup$
Row transformations retain the set of roots of system corresponding to the matrix, column transformations don't. However iff your aim is just calculating the determinant you can mix row and column transformations.
$endgroup$
– P Vanchinathan
Jan 18 at 2:54
$begingroup$
Row transformations retain the set of roots of system corresponding to the matrix, column transformations don't. However iff your aim is just calculating the determinant you can mix row and column transformations.
$endgroup$
– P Vanchinathan
Jan 18 at 2:54
$begingroup$
Using $det(A^T) =det A$, we see that the column operations change the determinant the same way as the row operations.
$endgroup$
– Berci
Jan 18 at 11:02
$begingroup$
Using $det(A^T) =det A$, we see that the column operations change the determinant the same way as the row operations.
$endgroup$
– Berci
Jan 18 at 11:02
add a comment |
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