1. ## Finding the determinant

Here is the link to the question, letter (b) is where I'm confused.

I'm confused about how they got det(A) = det(I) just from knowing what type of elementary operations were performed on the matrix. Thanks!

2. Originally Posted by pakman
Here is the link to the question, letter (b) is where I'm confused.

I'm confused about how they got det(A) = det(I) just from knowing what type of elementary operations were performed on the matrix. Thanks!

Once they have have it as a product they use the property of determinants

$\displaystyle det(AB)=det(A) \cdot det(B)$

since $\displaystyle E_3E_2E_1A=I \iff det(A)=det(E_1^{-1})det(E_2^{-1})det(E_3^{-1})$

so for the big conslusion if you swap two rows (columns ) in a matrix you get the negative of the determinant. If you multiply a row by a constnat it multiplies the determinant by the same constant (not zero)

Adding a multiple of a row (column) does not change the value of the determinant

$\displaystyle det(A)=\underbrace{det(E_1^{-1})}_{RowSwap=-1} \underbrace{det(E_2^{-1})}_{AddMultiple of a Row=1} \underbrace{det(E_3^{-1})}_{Add Multiple of A Row=1}=(-1)(1)(1)=-1$