| sinx cosx 1 |
| (sinx)^2 (cosx)^2 1 |
| (sinx)^3 (cosx)^3 1 |
Evaluate and factorise the determinant of the matrix.
for starters
s *c^2 + c*s^3 + c^3*s^2 - c^2s^3 - sc^3 -s^2 c
s c [ c + s^2 + c^2 s - c s^2 - c^2 - s ]
It's never going to get very simple, but one factorization is
1/4 (c-s)((c+s)^2 - 1)(c+s-1)^2
To evaluate the determinant of the matrix, we can use the properties of determinants. The determinant of a 3x3 matrix can be calculated by expanding along any row or column. Let's evaluate the determinant by expanding along the first column.
To calculate it, use the formula:
Determinant = a11 * C11 + a21 * C21 + a31 * C31
Where:
- a11, a21, and a31 are the entries in the first column of the matrix.
- C11, C21, and C31 are the corresponding cofactors.
Using this formula, we can evaluate the determinant as follows:
Determinant = sin(x) * C11 + sin(x)^2 * C21 + sin(x)^3 * C31
Now, let's find the cofactors C11, C21, and C31:
C11 = (-1)^(1+1) * (cos(x)^2 * 1 - 1 * (cos(x)^3)) = cos(x)^2 - cos(x)^3
C21 = (-1)^(2+1) * (sin(x)^2 * 1 - 1 * (sin(x)^3)) = -sin(x)^2 + sin(x)^3
C31 = (-1)^(3+1) * (sin(x)^2 * (cos(x)^3) - (sin(x)^3) * (cos(x)^2)) = sin(x)^2 * cos(x)^3 - sin(x)^3 * cos(x)^2
Now substitute the cofactors back into the determinant equation:
Determinant = sin(x) * (cos(x)^2 - cos(x)^3) + sin(x)^2 * (-sin(x)^2 + sin(x)^3) + sin(x)^3 * (sin(x)^2 * cos(x)^3 - sin(x)^3 * cos(x)^2)
Simplifying this expression further may be difficult, so it might be more appropriate to leave it in this expanded form.