Complex Numbers

To evaluate the determinant of the matrix:
|

‌ −bc a2	‌ c a	‌ b a
‌ c b	−‌ ac b2	‌ a b
‌ b c	‌ a c	−‌ ab c2

|
we can simplify the process by factoring out ‌

1

a2

,‌

1

b2

, and ‌

1

c2

from rows R1,R2, and R3 respectively. This allows us to focus on a simpler determinant:
⇒‌‌‌

1

a2b2c2

|

−bc	ac	ab
bc	−ac	ab
bc	ac	−ab

|
Next, apply row operations to simplify further: replace R2 with R2+R1 and R3 with R3+R1 :
⇒‌‌‌

1

a2b2c2

|

−bc	ac	ab
0	0	2ab
0	2ac	0

|
This matrix is now triangular, allowing us to compute the determinant by multiplying the diagonal elements:
⇒‌‌‌

1

a2b2c2

×(4a2b2c2)=4
Thus, the value of the determinant is 4 .