290 Expanding this determinant about the the second column....

[karush]

View attachment 8905
where does 32 (red) come from ?
nevermind looks its (-8)(4)=-32

but will probable have more ? on this example

 

[HOI]

Expanding on the second column we have
$\left|\begin{array}{ccc}\lambda- 5 & 0 & 4 \\ -12 & \lambda- 1 & 12 \\ -8 & 0 & \lambda- 7\end{array}\right|$$= -(\lambda- 1)\left|\begin{array}{cc} \lambda- 5 & 4 \\ -8 & \lambda+ 7 \end{array}\right|$$= -(\lambda- 1)[(\lambda- 5)(\lambda+ 7)- (-8)(4)]$$=-(\lambda- 1)[(\lambda- 5)(\lambda+ 7)+ 32]$.
The "32" comes from the -8 and 4 in the original matrix.

 

[karush]

thank you for helping
I am getting the impression not to many here can help with Linear Algebra
despite all the views of curiosity I did this with another problem but was kinda with it

https://www.physicsforums.com/attachments/8909