Prove the following determinant identities without expanding the determinants

Question

a) $$\begin{vmatrix} \sin^2 x & \cos^2 x & \cos 2x \\ \sin^2 y & \cos^2 y & \cos 2y \\ \sin^2 z & \cos^2 z & \cos 2z \\ \end{vmatrix} = 0;$$

$$\begin{vmatrix} \sin^2 x & \cos^2 x & \cos^2x-\sin^2x \\ \sin^2 y & \cos^2 y & \cos^2y-\sin^2y \\ \sin^2 z & \cos^2 z & \cos^2z-\sin^2z \\ \end{vmatrix} = 0;$$

$$\begin{vmatrix} \cos^2 x & \cos^2 x & \cos^2x-\sin^2x \\ \cos^2 y & \cos^2 y & \cos^2y-\sin^2y \\ \cos^2 z & \cos^2 z & \cos^2z-\sin^2z \\ \end{vmatrix} = 0;$$

b) $$\begin{vmatrix} 1 & a & p+c \\ 1 & p & c+a \\ 1 & c & a+p \\ \end{vmatrix} = 0,$$

$$\begin{vmatrix} 1 & a & 1\cdot(a+p+c) \\ 1 & p & 1\cdot(p+c+a) \\ 1 & c & 1\cdot(c+a+p) \end{vmatrix} = (a + p + c)\cdot \begin{vmatrix} 1 & a & 1 \\ 1 & p & 1 \\ 1 & c & 1 \end{vmatrix} = 0.$$

I am stuck on a), any hints and help is appreciate, and please check if b) is correct.

I've replaced the title with something more coherent -- One does not "prove" a determinant. (One could compute it, or prove that some claim about its value is correct.) — Andrés E. Caicedo, Oct 14 '13 at 23:46
b) is correct. You have made the two columns linearly dependent. — Ross Millikan, Oct 14 '13 at 23:54

score 5 · Accepted Answer · answered Oct 14 '13 at 23:40

5

Remember that $\cos 2x = \cos^2 x - \sin^2 x$.

answered Oct 14 '13 at 23:40

Smylic

6,715

Thanks a lot. I already looked up and found this, however, I'm unsure what operation shall I go next for. – user95523 Oct 14 '13 at 23:44
Futher you can add any column/row multiplied by any real number to any other column/row. – Smylic Oct 14 '13 at 23:49
2

The third column is a linear combination of the first two. – Ted Shifrin Oct 14 '13 at 23:49
I've edited the answer, could you check if I'm on the right track, if so what should I do next? – user95523 Oct 14 '13 at 23:59
Now you have two equal columns, so matrix is degenerate and determinant is zero. – Smylic Oct 15 '13 at 00:03

Prove the following determinant identities without expanding the determinants

1 Answers1