So expanding along third
column (C3), we get
∆ =
–1
3
1
2
1
2
4
– 0
0
4
1
4
1
–1
3
+
= 4 (–1 – 12) – 0 + 0  = – 52
Example 4 Evaluate ∆ = 
0
sin
–cos
–sin
0
sin
cos
–sin
0
α
α
α
β
α
β Rationalised 2023-24
DETERMINANTS     81
Solution Expanding along R1, we get
∆ =
0
sin
–sin
sin
–sin
0
0
– sin
– cos
–sin
0
cos
0
cos
–sin
β
α
β
α
α
α
β
α
α
β
= 0 – sin α (0 – sin β cos α) – cos α (sin α sin β – 0)
= sin α sin β cos α – cos α sin α sin β = 0
Example 5 Find values of x for which 3
3
2
1
4
1
x
x
= Solution We have 3
3
2
1
4
1
x
x
=
i e