Matrices and Determinants

To solve the given determinant equation, we start by simplifying the expression:
|

x−2	3(x−1)	5(x−1)
x−4	3(x−3)	5(x−5)
x−8	3(x−9)	5(x−25)

|=0
This simplifies to:
|

x−2	x−1	x−1
x−4	x−3	x−5
x−8	x−9	x−25

|=0
Next, perform column operations to simplify the determinant. Subtract the second column from the first:
C1⟶C1−C2⟹|

−1	x−1	x−1
−1	x−3	x−5
1	x−9	x−25

|=0
Now, apply row operations:
Add the third row to the first row: R1⟶R1+R3
Add the third row to the second row: R2⟶R2+R3
This results in:
|

0	2x−10	2x−26
0	2x−12	2x−30
1	x−9	x−25

|=0

To solve, expand the determinant by the first column:
(2x−30)(2x−10)−(2x−12)(2x−26)=0
This simplifies to:
2(x−15)⋅2(x−5)−2(x−6)⋅2(x−13)=0
Further simplifying, we have:
(x−15)(x−5)−(x−6)(x−13)=0
Solving this, we calculate:
x2−20x+75−(x2−19x+78)=0
This leads to:
−x−3=0⟹x=−3
Therefore, x=−3 satisfies the equation x2+2x−3=0.