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Math Help - Finding the determinant

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    Finding the determinant

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

    http://www.math.ohio-state.edu/~pwkim/math571/addpr.pdf

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

    http://www.math.ohio-state.edu/~pwkim/math571/addpr.pdf

    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

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

    since 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

     <br />
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<br />
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