PROPERTIES OF DETERMINANTS 1. Determinant of a Transpose The determinant of a transpose AT of A is equal to the determinant of A. det(A") = det(A) 2. Interchange of Rows and Columns The determinant changes its sign if two adjacent rows (or columns) are interchanged. ja1 a12 a21 az2 ** ain ** azn ja21 a22 a1 a12 aznl ..* annl |ani an2 *** an 3. Multiplication of a determinant by a Number k det(A) = det(A') Where: The matrix A' differs from A in that any one of its row or columns is multiplied by k. PROPERTIES OF DETERMINANTS 4. Determinant with equal rows or columns - The determinant of A is zero if two of its rows or columns are proportional to each other element by element. > The determinant of A is zero if two rows or columns are equal. The determinant of A is zero if a row or column has only null elements. 5. Sum of Determinants Consider matrix A = [a and matrix A', with all elements equal to A except for one row or column: ran a12 an] a2n a2 an ain azn A = A' = b bz Then: det(A) + det(A') = an + bu az + bi2 an + bin ... Lani ann lan an2 ann an2 ann
PROPERTIES OF DETERMINANTS 1. Determinant of a Transpose The determinant of a transpose AT of A is equal to the determinant of A. det(A") = det(A) 2. Interchange of Rows and Columns The determinant changes its sign if two adjacent rows (or columns) are interchanged. ja1 a12 a21 az2 ** ain ** azn ja21 a22 a1 a12 aznl ..* annl |ani an2 *** an 3. Multiplication of a determinant by a Number k det(A) = det(A') Where: The matrix A' differs from A in that any one of its row or columns is multiplied by k. PROPERTIES OF DETERMINANTS 4. Determinant with equal rows or columns - The determinant of A is zero if two of its rows or columns are proportional to each other element by element. > The determinant of A is zero if two rows or columns are equal. The determinant of A is zero if a row or column has only null elements. 5. Sum of Determinants Consider matrix A = [a and matrix A', with all elements equal to A except for one row or column: ran a12 an] a2n a2 an ain azn A = A' = b bz Then: det(A) + det(A') = an + bu az + bi2 an + bin ... Lani ann lan an2 ann an2 ann
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 7AEXP
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