Evaluate the following determinants : ​i. |(x,-7)(x,5x+1)| ii.|(cos θ,-sin θ)(sin θ, cos θ)| iii. iv. ​

Question

Evaluate the following determinants :

i. \(\begin{vmatrix} x &-7 \\[0.3em] x & 5x+1 \\[0.3em] \end{vmatrix}\)

ii. \(\begin{vmatrix} cos\,\theta &-sin\,\theta \\[0.3em] sin\,\theta & cos\,\theta \\[0.3em] \end{vmatrix}\)

iii. \(\begin{vmatrix} cos\,15° &-sin\,15° \\[0.3em] sin\,75° & cos\,75° \\[0.3em] \end{vmatrix}\)

iv. \(\begin{vmatrix} a+ib &c+id \\[0.3em] -c+id & a-ib \\[0.3em] \end{vmatrix}\)

Answer 1

i. Let A = \(\begin{vmatrix} x &-7 \\[0.3em] x & 5x+1 \\[0.3em] \end{vmatrix}\)

⇒ |A| = x(5x + 1) – (–7)x 

|A| = 5x² + 8x

ii. Let A = \(\begin{vmatrix} cos\,\theta &-sin\,\theta \\[0.3em] sin\,\theta & cos\,\theta \\[0.3em] \end{vmatrix}\)

⇒ |A| = cosθ × cosθ – (–sinθ) x sinθ 

|A| = cos 2θ + sin 2θ 

|A| = 1

iii. Let A = \(\begin{vmatrix} cos\,15° &-sin\,15° \\[0.3em] sin\,75° & cos\,75° \\[0.3em] \end{vmatrix}\)

⇒ |A| = cos15° × cos75° + sin15° x sin75° 

|A| = cos(75 – 15)° 

|A| = cos60° 

|A| = 0.5.

iv. A = \(\begin{vmatrix} a+ib &c+id \\[0.3em] -c+id & a-ib \\[0.3em] \end{vmatrix}\)

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

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

= a² – i² b² + c² – i² d² = a² – (–1)b² + c² – (–1)d² 

= a² + b² + c² + d²