Evaluate the following determinants :i. \(\begin{vmatrix}x &-7 \\[0.3

1.	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» 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

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»

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