Matrix Expressions : Factorizing

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1st equation:

Assuming x is not zero, we do
R1/x+ R2 ---> R2. Therefore, R2= [0 x 0 -10- 8/x] and that does not change the determinant.

Then we do R2/x+ R3 ---> R3. Therefore, R3= [0 0 x -1- 10/x- 8/x^2] and again that does not change the determinant.

Finally R3/x+ R4 ---> R4. Therefore, R4= [0 0 0 1- 1/x- 10/x^2- 8/x^3]. This one also does not change the determinant, so all in all we have:

|x 0 0 -8 |
|0 x 0 -10- 8/x |
|0 0 x -1 -10/x -8/x^2 |= 0
|0 0 0 1- 1/x- 10/x^2- 8/x^3|

Now the matrix is upper-triangular and its determinant is the multiplication of the components on its main diagonal. That means:

x^3(1- 1/x- 10/x^2- 8/x^3)= 0, or
x^3- x^2- 10x- 8= 0 ---> We can see (using trial and error) that x= -1 is a root. The other roots can be found if we divide ...