With every square matrix A of a certain order, we associate a number called the determinant of A and it is denoted byA or det A. If a matrix is not square, we can not associate a determinant with it.
Properties: (1) The value of the determinant remains unaltered if the rows and columns are interchanged.
(2) If two adjacent rows (or columns) of a determinant are interchanged, the numerical value remains same but the sign of the determinant is changed.
(3) If two rows (or columns) of a determinant are identical, the value of the determinant is zero.
(4) If all the elements of any row (or column) are multiplied by the same constant, then the original determinant is multiplied by that constant.
(5) If the elements of any row (or column) of a determinant are multiplied in order by the cofactors of the corresponding elements of any other row (or column), then the sum of the products thus obtained is zero.
(6) If each element of any row (or column) is the sum of two numbers, then the determinant can be expressed as the sum of the determinants whose other rows (or columns) are not altered.
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