# Determinants

## In algebra, the determinant is the numerical value that can be assigned to a matrix of order n x n, or, we can say, to a square matrix. The determinant of a square matrix A is denoted as det(A), det A, or |A|. Usually the |A| is used to denote the determinant of a square matrix A. The determinant of a matrix A = [a11]1x1, is given by |a11|. The determinant of a matrix of order 2x2 is (a11a22 – a12a21). The determinant of a matrix of order 3x3 is {a11(a33a22 – a23a32) – a12(a21a33 – a23a31) + a13(a21a32 – a22a31)} . System of Linear Equation can be represented in Matrix form. Properties of the determinant of a square matrix: 1. | A’| = | A |, where A’ = Transpose of A. 2. If we interchange any two rows (or columns), then sign of determinant changes. 3. If any two or any two columns are identical or proportional, the value of determinant is zero. 4. If we multiply each element of a row or a column of a determinant by constant k, then value of determinant is multiplied by k. 5. Multiplying a determinant by k means multiplying elements of only one row (or one column) by k. 6. If A = [aij]3 x 3, then | k.A | = k3| A |. 7. If elements of a row or a column in a determinant can be expressed as sum of two or more elements, then the given determinant can be expressed as sum of two or more determinants. 8. If equal multiples of each element of a row (or column) of a determinant are added to the corresponding elements of other row (or column), then the value of the determinant does not change. Keywords: Determinants; Properties of Determinants; Area of Triangle; Minors and Cofactors; Adjoint and Inverse of a Matrix; Applications of Determinants and Matrices

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