Question

Evaluate the following determinants: `|(1, "i", 3),("i"^3, 2, 5),(3, 2, "i"^4)|`

Answer 1

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

= `|(1, "i", 3),(-"i", 2, 5),(3, 2, 1)|`     ...[∵ i² = – 1, i⁴ = 1]

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

= 1(2 – 10) – i(– i –15) + 3(– 2i – 6)
= – 8 + i² + 15i – 6i – 18
= i² – 26 + 9i
= – 1 – 26 + 9i          ...[∵ i² = – 1]
= – 27 + 9i