Properties of Determinants
- The value of a determinants does not change when rows and columns are interchanged. i.e., ⏐AT⏐= ⏐A⏐
- If any row (or column) of a matrix A is completely zero, then ⏐A⏐= 0. Such a row (or column) is called a zero row (or column).
Also, if any two rows (or columns) of a matrix A are identical, then ⏐A⏐ = 0. - If any two rows or two columns of a determinant are interchanged, the value of the determinant is multiplied by –1.
- If all elements of the one row (or one column) of a determinant are multiplied by the same number k, the value of the determinant is k times the value of the given determinant.
- If A be n-rowed square matrix, and k be any scalar, then |kA| = kn|A|
- (a) In a determinant the sum of the products of the elements of any row (or column) with the cofactors of corresponding elements of any row or column is equal to the determinant value.
(b) In the determinant, the sum of the products of the elements of any row (or column) with the cofactors of some other row or column is zero. - If to the elements of a row (or column) of a determinant are added m times the corresponding elements of another row (or column), the value of the determinant thus obtained is equal to the value of the original determinant.
- ⏐AB⏐ = ⏐A⏐*⏐B⏐ And based on this, we can prove the following:
(a) ⏐An⏐ = (⏐A⏐)n
(b) ⏐A–1⏐ = 1/ |A| - Using the fact that A . Adj A = ⏐A⏐ . I, the following can be proved for An × n.
- The determinant of any n × n matrix whose all principle diagonals are the same (say a) and all other remaining entries are the same (say b) is (a + 3b) (a – b) (a – b) (a – b) = (a + 3b) (a – b)³.
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