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    Proof about determinants

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    From Determinants.

    (From Determinants. Prove the proposition without using Cofactors/Cramer's rule.)

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    © BrainMass Inc. brainmass.com December 24, 2021, 7:19 pm ad1c9bdddf
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    Proof:
    We should use the following properties of determinant.
    1. Switching two rows or columns changes the sign.
    2. Scalar multiplication of a row by a constant multiplies the determinant by .
    3. Multiplies of rows and columns can be added together without changing the determinant's value.

    Now we consider the three types of elementary matrices.
    Type 1: exchanges row i and row j, then exchanges row i and row j in . From the property 1, we have . On the other hand, we note that , then we have . Thus .
    Type 2: multiplies row i by scalar , then multiplies row i of by . From the property 2, we have . On the other hand, we note that , then we have . Thus .
    Type 3: adds the scalar multiple of row i to row j, then adds the scalar multiple of row i of to row j of . From the property 3, we have . On the other hand, , then . Thus .
    Now for a general elementary matrix , is a multiplication of the above three types. Thus we have

    Therefore, we always have . Done.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 7:19 pm ad1c9bdddf>
    https://brainmass.com/math/matrices/proof-about-determinants-179016

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