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The basic theorems used in determining equality using determinants are the associative property, the multiplicative property, and the transitive property.
The associative property states that the order in which matrices are multiplied does not affect the result. This allows us to manipulate the order of the matrices in the determinant without changing the final value, making it easier to prove equality.
The multiplicative property states that multiplying a row or column of a determinant by a constant also multiplies the value of the determinant by that constant. This allows us to manipulate the values within the determinant without changing the final result, making it easier to prove equality.
Yes, the transitive property states that if A = B and B = C, then A = C. This means that if we can prove that two determinants are equal to a third determinant, we can use the transitive property to prove that the first two determinants are also equal.
The process for proving equality using determinants involves manipulating the determinants using the basic theorems until they are equal in their final form. This may involve multiplying rows or columns by constants, manipulating the order of the matrices, or using the transitive property to compare multiple determinants. By following these steps, we can prove that two determinants are equal.
