Spencer asked in Science & MathematicsMathematics · 6 months ago

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
• 6 months ago

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.)