1. ## Determinant and eigenvalues

I am trying to figure out different ways og getting the eigenvalues so i will be able to do it quickly in an exam. I am wondering if all my reasoning is correct because i am still making a few mistakes.

1)row operations do not affect the determinant if we multiply be -1, each time we change a row.

2)if we divide a row by a constant what do i do to get the right answer for the determinant?

3) column operations do not affect the determinant

2. Originally Posted by ulysses123
I am trying to figure out different ways og getting the eigenvalues so i will be able to do it quickly in an exam. I am wondering if all my reasoning is correct because i am still making a few mistakes.

1)row operations do not affect the determinant if we multiply be -1, each time we change a row.
I have no idea what you mean by "change a row". If you swap two rows in a determinant, then that multiplies the determinant by -1. So if you swap two rows and multiply by -1, that does not change the determinant.

2)if we divide a row by a constant what do i do to get the right answer for the determinant?
Dividing (multiplying) a row by a constant multiplies (divides) the entire determinant by that constant so to keep the same determinant, do the opposite: if you multiply a row by a constant and divide the determinant by the same constant you do not change the determinant.

3) column operations do not affect the determinant
No, "column operations" do exactly the same thing as row operations:

Multiplying a row or column by a number multiplies the entire determinant by that number.

Swapping two rows or columns multiplies the entire determinant by -1.

Adding a multiple of one row or column to another does not change the value of the determinant.