| Determinants |
Determinants :
➤ In linear algebra, a determinant is a special number that can be determined from a square matrix.
➤ A number associated with a square matrix of order ‘n’ is known as the determinant of the given matrix. This number can be a real number as well as a complex number depending upon the entries of the square matrix.
➤ We denote the determinant of any matrix A by det (A), det A, or |A|.
➤ In a determinant, the number of rows should be equal to the number of columns.
| Properties of Determinants |
Properties of Determinants :
➤ If all the elements of a row (or column) are zero, then the determinant is zero.
➤ If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.
➤ The sign of a determinant will change when we interchange any two rows or columns of a determinant with each other.
➤ In a triangular Matrix, the Determinant is equal to the product of the diagonal elements.
➤ If the Matrix XT is the transpose of Matrix X, then det (XT) = det (X)
➤ If all the elements of a row or columns in a determinant are expressed as a summation of two or more numbers, then the determinant can be broken down as a sum of corresponding smaller determinants.
➤ If any pair of rows or columns of a determinant are exactly identical or proportion by same amount, then the determinant is zero.
➤ For an identity matrix, the determinant equals to 1. (An identity matrix is a matrix in which the main diagonal consists of all 1 elements and the rest elements are 0.)
➤ The determinant of a triangular matrix equals the product of the main diagonal elements. (A triangular matrix is a matrix in which either the elements above the main diagonal are 0 or the elements below the main diagonal are 0.)
| Examples |
Example 1 :
Find the determinant of the 2 ✕ 2 matrix below.
Example 2 :
Find the determinant of the 3 ✕ 3 matrix below.
