Mixing
Determinant of a matrix whose elements are trigonometric functions
$begingroup$
Calculate:
$$detbegin{pmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
cos2varphi & sin2varphi & 2cos2varphi & 2sin2varphi \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{pmatrix}
$$
By elementary row operation,it is equivalent to calculate
$$detbegin{pmatrix}
cosvarphi & sinvarphi & 0 & 0 \
0 & 0 & cos2varphi & sin2varphi \
-cos3varphi & -sin3varphi & 2cos3varphi & 2sin3varphi \
-2cos4varphi & -2sin4varphi & 3cos4varphi & 3sin4varphi
end{pmatrix}
$$
Though this form is a bit more convenient for Laplace Expansion, it still requires a lot of effort to obtain the result.
linear-algebra determinant
edited Jan 27 at 11:49
Chinnapparaj R
5,8082928
asked Jan 27 at 11:25
Tao XTao X
865
$endgroup$
add a comment |
$begingroup$
Calculate:
$$detbegin{pmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
cos2varphi & sin2varphi & 2cos2varphi & 2sin2varphi \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{pmatrix}
$$
By elementary row operation,it is equivalent to calculate
$$detbegin{pmatrix}
cosvarphi & sinvarphi & 0 & 0 \
0 & 0 & cos2varphi & sin2varphi \
-cos3varphi & -sin3varphi & 2cos3varphi & 2sin3varphi \
-2cos4varphi & -2sin4varphi & 3cos4varphi & 3sin4varphi
end{pmatrix}
$$
Though this form is a bit more convenient for Laplace Expansion, it still requires a lot of effort to obtain the result.
linear-algebra determinant
edited Jan 27 at 11:49
Chinnapparaj R
5,8082928
asked Jan 27 at 11:25
Tao XTao X
865
$endgroup$
$begingroup$
I would not say it is a lot of effort; knowing a few basic trigonometric identities makes it a few minutes worth of work at most. Did you try anything further?
$endgroup$
– Servaes
Jan 27 at 11:30
add a comment |
$begingroup$
Calculate:
$$detbegin{pmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
cos2varphi & sin2varphi & 2cos2varphi & 2sin2varphi \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{pmatrix}
$$
By elementary row operation,it is equivalent to calculate
$$detbegin{pmatrix}
cosvarphi & sinvarphi & 0 & 0 \
0 & 0 & cos2varphi & sin2varphi \
-cos3varphi & -sin3varphi & 2cos3varphi & 2sin3varphi \
-2cos4varphi & -2sin4varphi & 3cos4varphi & 3sin4varphi
end{pmatrix}
$$
Though this form is a bit more convenient for Laplace Expansion, it still requires a lot of effort to obtain the result.
linear-algebra determinant
edited Jan 27 at 11:49
Chinnapparaj R
5,8082928
asked Jan 27 at 11:25
Tao XTao X
865
$endgroup$
Calculate:
$$detbegin{pmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
cos2varphi & sin2varphi & 2cos2varphi & 2sin2varphi \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{pmatrix}
$$
By elementary row operation,it is equivalent to calculate
$$detbegin{pmatrix}
cosvarphi & sinvarphi & 0 & 0 \
0 & 0 & cos2varphi & sin2varphi \
-cos3varphi & -sin3varphi & 2cos3varphi & 2sin3varphi \
-2cos4varphi & -2sin4varphi & 3cos4varphi & 3sin4varphi
end{pmatrix}
$$
Though this form is a bit more convenient for Laplace Expansion, it still requires a lot of effort to obtain the result.
linear-algebra determinant
linear-algebra determinant
edited Jan 27 at 11:49
Chinnapparaj R
5,8082928
asked Jan 27 at 11:25
Tao XTao X
865
edited Jan 27 at 11:49
Chinnapparaj R
5,8082928
asked Jan 27 at 11:25
Tao XTao X
865
edited Jan 27 at 11:49
Chinnapparaj R
5,8082928
edited Jan 27 at 11:49
Chinnapparaj R
5,8082928
edited Jan 27 at 11:49
Chinnapparaj R
5,8082928
5,8082928
asked Jan 27 at 11:25
Tao XTao X
865
asked Jan 27 at 11:25
Tao XTao X
865
asked Jan 27 at 11:25
Tao XTao X
865
865
$begingroup$
I would not say it is a lot of effort; knowing a few basic trigonometric identities makes it a few minutes worth of work at most. Did you try anything further?
$endgroup$
– Servaes
Jan 27 at 11:30
add a comment |
$begingroup$
I would not say it is a lot of effort; knowing a few basic trigonometric identities makes it a few minutes worth of work at most. Did you try anything further?
$endgroup$
– Servaes
Jan 27 at 11:30
$begingroup$
I would not say it is a lot of effort; knowing a few basic trigonometric identities makes it a few minutes worth of work at most. Did you try anything further?
$endgroup$
– Servaes
Jan 27 at 11:30
$begingroup$
I would not say it is a lot of effort; knowing a few basic trigonometric identities makes it a few minutes worth of work at most. Did you try anything further?
$endgroup$
– Servaes
Jan 27 at 11:30
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Here's one way to do it:
begin{align*}
& begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
cos2varphi & sin2varphi & 2cos2varphi & 2sin2varphi \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 2sinvarphi & 2cos3varphi & 3sin3varphi - frac{cos3varphi sinvarphi}{cosvarphi} \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 2sinvarphi & 2cos3varphi & 3sin3varphi - frac{cos3varphi sinvarphi}{cosvarphi} \
0 & frac{sin3varphi}{cosvarphi} & 3cos4varphi & 4sin4varphi - cos(4varphi)tan(x)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & frac{sin3varphi}{cosvarphi} & 3cos4varphi & 4sin4varphi - cos(4varphi)tan(x)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & 0 & -2 sin^2varphi (3 + 4 cos2varphi) & sin2varphi (4 cos2varphi - 1)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & 0 &0 & -tanvarphi
end{vmatrix} \
= & 4 tan^2varphi cos^2varphi sin^2varphi
= 4 sin^4varphi.
end{align*}
While it looks very complicated, notices that all terms of rows 2 - 4, which are above the diagonal, don't really matter, so don't have to be computed explicitly.
answered Jan 27 at 11:46
Viktor GlombikViktor Glombik
1,2372528
$endgroup$
add a comment |
Your Answer
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1 Answer
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1 Answer
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active
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active
oldest
votes
$begingroup$
Here's one way to do it:
begin{align*}
& begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
cos2varphi & sin2varphi & 2cos2varphi & 2sin2varphi \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 2sinvarphi & 2cos3varphi & 3sin3varphi - frac{cos3varphi sinvarphi}{cosvarphi} \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 2sinvarphi & 2cos3varphi & 3sin3varphi - frac{cos3varphi sinvarphi}{cosvarphi} \
0 & frac{sin3varphi}{cosvarphi} & 3cos4varphi & 4sin4varphi - cos(4varphi)tan(x)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & frac{sin3varphi}{cosvarphi} & 3cos4varphi & 4sin4varphi - cos(4varphi)tan(x)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & 0 & -2 sin^2varphi (3 + 4 cos2varphi) & sin2varphi (4 cos2varphi - 1)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & 0 &0 & -tanvarphi
end{vmatrix} \
= & 4 tan^2varphi cos^2varphi sin^2varphi
= 4 sin^4varphi.
end{align*}
While it looks very complicated, notices that all terms of rows 2 - 4, which are above the diagonal, don't really matter, so don't have to be computed explicitly.
answered Jan 27 at 11:46
Viktor GlombikViktor Glombik
1,2372528
$endgroup$
add a comment |
$begingroup$
Here's one way to do it:
begin{align*}
& begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
cos2varphi & sin2varphi & 2cos2varphi & 2sin2varphi \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 2sinvarphi & 2cos3varphi & 3sin3varphi - frac{cos3varphi sinvarphi}{cosvarphi} \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 2sinvarphi & 2cos3varphi & 3sin3varphi - frac{cos3varphi sinvarphi}{cosvarphi} \
0 & frac{sin3varphi}{cosvarphi} & 3cos4varphi & 4sin4varphi - cos(4varphi)tan(x)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & frac{sin3varphi}{cosvarphi} & 3cos4varphi & 4sin4varphi - cos(4varphi)tan(x)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & 0 & -2 sin^2varphi (3 + 4 cos2varphi) & sin2varphi (4 cos2varphi - 1)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & 0 &0 & -tanvarphi
end{vmatrix} \
= & 4 tan^2varphi cos^2varphi sin^2varphi
= 4 sin^4varphi.
end{align*}
While it looks very complicated, notices that all terms of rows 2 - 4, which are above the diagonal, don't really matter, so don't have to be computed explicitly.
answered Jan 27 at 11:46
Viktor GlombikViktor Glombik
1,2372528
$endgroup$
add a comment |
$begingroup$
Here's one way to do it:
begin{align*}
& begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
cos2varphi & sin2varphi & 2cos2varphi & 2sin2varphi \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 2sinvarphi & 2cos3varphi & 3sin3varphi - frac{cos3varphi sinvarphi}{cosvarphi} \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 2sinvarphi & 2cos3varphi & 3sin3varphi - frac{cos3varphi sinvarphi}{cosvarphi} \
0 & frac{sin3varphi}{cosvarphi} & 3cos4varphi & 4sin4varphi - cos(4varphi)tan(x)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & frac{sin3varphi}{cosvarphi} & 3cos4varphi & 4sin4varphi - cos(4varphi)tan(x)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & 0 & -2 sin^2varphi (3 + 4 cos2varphi) & sin2varphi (4 cos2varphi - 1)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & 0 &0 & -tanvarphi
end{vmatrix} \
= & 4 tan^2varphi cos^2varphi sin^2varphi
= 4 sin^4varphi.
end{align*}
While it looks very complicated, notices that all terms of rows 2 - 4, which are above the diagonal, don't really matter, so don't have to be computed explicitly.
answered Jan 27 at 11:46
Viktor GlombikViktor Glombik
1,2372528
$endgroup$
Here's one way to do it:
begin{align*}
& begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
cos2varphi & sin2varphi & 2cos2varphi & 2sin2varphi \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
cos3varphi & sin3varphi & 3cos3varphi & 3sin3varphi \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 2sinvarphi & 2cos3varphi & 3sin3varphi - frac{cos3varphi sinvarphi}{cosvarphi} \
cos4varphi & sin4varphi & 4cos4varphi & 4sin4varphi
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 2sinvarphi & 2cos3varphi & 3sin3varphi - frac{cos3varphi sinvarphi}{cosvarphi} \
0 & frac{sin3varphi}{cosvarphi} & 3cos4varphi & 4sin4varphi - cos(4varphi)tan(x)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & frac{sin3varphi}{cosvarphi} & 3cos4varphi & 4sin4varphi - cos(4varphi)tan(x)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & 0 & -2 sin^2varphi (3 + 4 cos2varphi) & sin2varphi (4 cos2varphi - 1)
end{vmatrix} \
= & begin{vmatrix}
cosvarphi & sinvarphi & cosvarphi & sinvarphi \
0 & tanvarphi & cos2varphi & 2sin2varphi - frac{sinvarphicos2varphi}{cosvarphi} \
0 & 0 & -4cosvarphisin^2varphi & sin3varphi - sinvarphi \
0 & 0 &0 & -tanvarphi
end{vmatrix} \
= & 4 tan^2varphi cos^2varphi sin^2varphi
= 4 sin^4varphi.
end{align*}
While it looks very complicated, notices that all terms of rows 2 - 4, which are above the diagonal, don't really matter, so don't have to be computed explicitly.
answered Jan 27 at 11:46
Viktor GlombikViktor Glombik
1,2372528
edited Jan 27 at 11:51
answered Jan 27 at 11:46
Viktor GlombikViktor Glombik
1,2372528
answered Jan 27 at 11:46
Viktor GlombikViktor Glombik
1,2372528
answered Jan 27 at 11:46
Viktor GlombikViktor Glombik
1,2372528
1,2372528
add a comment |
add a comment |
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$begingroup$
I would not say it is a lot of effort; knowing a few basic trigonometric identities makes it a few minutes worth of work at most. Did you try anything further?
$endgroup$
– Servaes
Jan 27 at 11:30