Among the statements: I: If vmatrix1&& &1& &&1vmatrix=0, then ^2+^2+^2=1, and II: If vmatr…

Question

Among the statements:

I: If $\begin{vmatrix}1&\cos\alpha&\cos\beta\ \cos\alpha&1&\cos\gamma\ \cos\beta&\cos\gamma&1\end{vmatrix}=0$, then $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$, and

II: If $\begin{vmatrix}x^2+x&x+1&x-2\ 2x^2+3x-1&3x&3x-3\ x^2+2x+3&2x-1&2x-1\end{vmatrix}=px+q$, then $p^2=196q^2$,

which of the following is correct?

Accepted Answer

Statement I: Expanding the determinant gives
$$1-\cos^2\gamma-\cos^2\alpha-\cos^2\beta+2\cos\alpha\cos\beta\cos\gamma=0,$$
i.e. $\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1+2\cos\alpha\cos\beta\cos\gamma$, not simply $1$ in general. So Statement I is false.

Statement II: Evaluating the determinant (by row operations) gives an expression of the form $px+q$ with the constant and linear coefficients computed as $q=-12$ and $-p+q=-36\Rightarrow p=24$. Then $p^2=576$ while $196q^2=196(144)=28224$, so $p^2\ne196q^2$. So Statement II is false.

Hence both statements are false.

Solutions · 1