## Properties Of Determinants

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# Properties of Determinants

We now state some useful properties of determinants of order three only. However, these properties hold for determinants of any order. These properties help a good deal in the evaluation of determinants. We use the notations Ri and Cj to denote respectively the ith row and the jth column of a determinant.

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**Property.**The value of a determinant remains unchanged if rows are changed into columns and columns into rows. That is,**Remark.**In terms of matrices, if A is square matrix, then |A| = |A’|.**Property.**If any two adjacent rows (or columns) of a determinant are interchanged, the value of the determinant so obtained is the negative of the value of the original determinant. That is,**Remark.**The notation R_{i }→ k R_{j }(C_{i }→ C_{j}) used to represent interchange of ith and jth row (column).**Property.**If any two rows (or columns) of a determinant are identical, the value of the determinant is zero. That is,**Property.**If each element of a row (or column) of a determinant is multiplied by a constant by a constant k, the value of the determinant so obtained is k times the value of the original determinant. That is,**Remark.**The notation Ri → k Ri (Ci → k Ci) is used to represent multiplication of each element of ith row (column) by the constant k.**Property.**If to the element of a row (or column) of a determinant are added k times the elements of another row (or column), the value of the determinant so obtained is equal to the value of the original determinant. That is,**Property.**If each element of a row (or column) of a determinant is the sum of two element, the determinant can be expressed as the sum of two determinant. That is,For more help in

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