Evaluate the following determinants:(i) \(\begin{vmatrix} x& -7 \\[0.

1.	Evaluate the following determinants:(i) \(\begin{vmatrix} x& -7 \\[0.3em] x & 5x + 1 \end{vmatrix}\)(ii) \(\begin{vmatrix} cos θ& -sin θ\\[0.3em]sin θ& cos θ\end{vmatrix}\)(iii) \(\begin{vmatrix} cos 15°& sin 15°\\[0.3em]sin 75°& cos 75°\end{vmatrix}\)(iv) \(\begin{vmatrix} a + ib & c + id\\[0.3em]-c + id& a - ib\end{vmatrix}\)
Answer» (i) \|A\| = x (5x + 1) – (–7) x \|A\| = 5x² + 8x (ii) \|A\| = cos θ × cos θ – (–sin θ) x sin θ \|A\| = cos²θ + sin²θ As we know that cos²θ + sin²θ = 1 \|A\| = 1 (iii) \|A\| = cos15° × cos75° + sin15° x sin75° As we know that cos (A – B) = cos A cos B + Sin A sin B On substituting this we get, \|A\| = cos (75 – 15)° \|A\| = cos60° \|A\| = 0.5 (iv) \|A\| = (a + ib) (a – ib) – (c + id) (–c + id) = (a + ib) (a – ib) + (c + id) (c – id) = a² – i² b² + c² – i² d² As we know that i² = -1 = a² – (–1) b² + c² – (–1) d² = a² + b² + c² + d²

Evaluate the following determinants:(i) \(\begin{vmatrix} x& -7 \\[0.3em] x & 5x + 1 \end{vmatrix}\)(ii) \(\begin{vmatrix} cos θ& -sin θ\\[0.3em]sin θ& cos θ\end{vmatrix}\)(iii) \(\begin{vmatrix} cos 15°& sin 15°\\[0.3em]sin 75°& cos 75°\end{vmatrix}\)(iv) \(\begin{vmatrix} a + ib & c + id\\[0.3em]-c + id& a - ib\end{vmatrix}\)

Answer»

(i) |A| = x (5x + 1) – (–7) x

|A| = 5x² + 8x

(ii) |A| = cos θ × cos θ – (–sin θ) x sin θ

|A| = cos²θ + sin²θ

As we know that cos²θ + sin²θ = 1

|A| = 1

(iii) |A| = cos15° × cos75° + sin15° x sin75°

As we know that cos (A – B) = cos A cos B + Sin A sin B

On substituting this we get, |A| = cos (75 – 15)°

|A| = cos60°

|A| = 0.5

(iv) |A| = (a + ib) (a – ib) – (c + id) (–c + id)

= (a + ib) (a – ib) + (c + id) (c – id)

= a² – i² b² + c² – i² d²

As we know that i² = -1

= a² – (–1) b² + c² – (–1) d²

= a² + b² + c² + d²