# How does this change the determinant of the Matrix?

If I have the matrix (M) with the determinant det(M), and then i have the matrix M' which is:

[            |0,0]

[   M      |0,0]

[            |0,0]

[0, 0, t-1|,-1,0]

[0, 0, t-1|,-1,t]

would you have to take into consideration the [t-1]s? since they are directly "under" the M matrix. I put this into my calculator using placeholder numbers, and I got the determinant of M to be 9, and M' to be -9, when i substituted a -1 diagonally across from the original M matrix, even though I also added terms below the matrix. So although I'm confident that it doesn't change the determinant, I'm wondering WHY it doesn't.

Relevance

Find the determinant of M' by using cofactor expansion on the last column of M'

You will get  t times the determinant of the matrix below.

[            |0]

[   M      |0]

[            |0]

[0, 0, t-1|,-1]

Now use cofactor expansion on the last column again.

The determinant of M' will be -t*det(M)

• Anonymous
6 months ago

Question is not clear,  But assuming det(M) is not zero, then changing the value of t *does* change det(M').

(If det(M) = 0 then det(M') = 0 also.)