Question

Evaluate the determinants in Exercises 1 and 2.
(i) $\begin{vmatrix}\cos\theta&-\sin\theta\\\\\sin\theta&\cos\theta\end{vmatrix}$

(ii) $\begin{vmatrix}x^2&-x+1&x-1\\\& x+1&x+1\end{vmatrix}$

Answer 1

(i) $\begin{vmatrix}\cos\theta&-\sin\theta\\\\\sin\theta&\cos\theta\end{vmatrix}$
=(cosθ)(cosθ)-(-sinθ)(sinθ)
=cos2 θ+ sin2 θ = 1

---

(ii) $\begin{vmatrix}x^2&-x+1&x-1\\\& x+1&x+1\end{vmatrix}$
= (x2− x + 1)(x + 1) − (x − 1)(x + 1)
= x3− x2+ x + x2 − x + 1 − (x2 − 1)
= x3 + 1 − 2+ 1
= x3 − x2+ 2