Question

Find x and y if `|(4i, i^3, 2i),(1, 3i^2, 4),(5, -3, i)|` = x + iy where i² = –1

Answer 1

`|(4i, i^3, 2i),(1, 3i^2, 4),(5, -3, i)|`  

`=|(4i, i^3, 2i),(1, 3i^2, 4),(5, -3, i)|`         ...[∵ i² = −1]

`= |(4i,-i,2i),(1,-3,4),(5,-3,i)|`

`= 4i|(-3, 4),(-3, i)|-(-i)|(1, 4),(5, i)| + 2i|(1, -3),(5, -3)|`

`= |(-3,4),(-3,i)| = (-3)(i)-(4)(-3)=-3i+12`

4i ⋅ (−3i + 12) = 4i ⋅ 12 − 4i ⋅ 3i = 48i − 12i² = 48i + 12

`|(1,4),(5,i)| = (1)(i) - (4)(5) = i-20`

+i ⋅ (i − 20) = i2 − 20i = −1 − 20i

`|(1,-3),(5,-3)| = (1)(-3) - (-3)(5) = -3+15 = 12`

2i ⋅ 12 = 24i

(48i + 12) + (−1 − 20i) + 24i = (48i − 20i + 24i) + (12 − 1) = 52i + 11

x = 11, y = 52