Finding the Determinant to find out if the matrix is invertible

[Sunwoo Bae]

Homework Statement

I am given the following matrix. I solved the question and found out that the resulting determinant is 0, thus the matrix is not invertible. However, I tried the question in another method, and I am getting a different answer for the determinant. Why is the second method not working?

Relevant Equations

Theorem: if a multiple of one row of A is added to another row to produce matrix B, then det B = det A

question:

My first attempt:

my second attempt:

So I am getting 0 (the right answer) for the first method and 40 for the second method. According to the theorem, shouldn't the determinant of the matrix remain the same when the multiple of one row is added to another row? Can anyone explain why the second method is not working?

Thank you very much!

 

[Mark44]

Homework Statement:: I am given the following matrix. I solved the question and found out that the resulting determinant is 0, thus the matrix is not invertible. However, I tried the question in another method, and I am getting a different answer for the determinant. Why is the second method not working?
Relevant Equations:: Theorem: if a multiple of one row of A is added to another row to produce matrix B, then det B = det A

question:
View attachment 266772

My first attempt:
View attachment 266773

my second attempt:
View attachment 266774So I am getting 0 (the right answer) for the first method and 40 for the second method. According to the theorem, shouldn't the determinant of the matrix remain the same when the multiple of one row is added to another row? Can anyone explain why the second method is not working?

Thank you very much!

You made a mistake in your second attempt, in the 4th row above the bottom. Take another look at the 3rd deteriminant. You have (-8)(-1) = 9.

 

[Sunwoo Bae]

You made a mistake in your second attempt, in the 4th row above the bottom. Take another look at the 3rd deteriminant. You have (-8)(-1) = 9.

I got it! Thank you so much!