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From Determinants.

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

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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.

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