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3. State the 3 2 2 6 5-8 8-4 appropriate theorems that justifies the given equality. 1 3 0

3. State the 3 2 2 6 5-8 8-4 appropriate theorems that justifies the given equality. 1 3 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Properties of Determinants
State the appropriate theorems that justifies the given equality, with solution. 


3. State the 3 2 2 6 5 - 8 appropriate theorems that justifies the given equality. 1 3 - 4 = 0
Transcribed Image Text:3. State the 3 2 2 6 5 - 8 appropriate theorems that justifies the given equality. 1 3 - 4 = 0
There are properties of determinants to simplify its computation. Theorem: Determinant of a Transpose If AT is the transpose of the n x n matrix A, then det AT = det A. Theorem: Two Identical Rows If any two rows (columns) of an n x n matrix A are the same, then det A = 0. Theorem: Zero Row or Column If all the entries in a row (column) of an n x n matrix A are zero, then det A = 0. Theorem: Interchanging Rows If B is the matrix obtained by interchanging any two rows (columns) of an n x n matrix A, then det B = -det A. Theorem: Constant Multiple of a Row If B is the matrix obtained from an n x n matrix A by multiplying a row (column) by a non zero real number k, then det B = k det A. Theorem: Determinant of a Matrix Product If A and B are both n x n matrices, then det AB = det A x det B Theorem: Determinant is unchanged If matrix B is obtained from multiplying entries from row(column) and adding corresponding entries to another row then the det B = det A. Theorem: Determinant of a triangular matrix If matrix A is an n x n triangular matrix (upper or lower). Then det A = a₁1₁a22...ann where a11a22...ann are the entries on the main diagonal of A. Theorem: A Property of Cofactors Suppose A is an n x n matrix. If we get the cofactors of the entries in the kth row. When we obtain the cofactor for the jth column a₁1Ck1 + a2Ck2 +...+ ainCkn = 0, for i #k a1jC1k + a2jC2k +...+ anjCnk = 0, for j #k
Transcribed Image Text:There are properties of determinants to simplify its computation. Theorem: Determinant of a Transpose If AT is the transpose of the n x n matrix A, then det AT = det A. Theorem: Two Identical Rows If any two rows (columns) of an n x n matrix A are the same, then det A = 0. Theorem: Zero Row or Column If all the entries in a row (column) of an n x n matrix A are zero, then det A = 0. Theorem: Interchanging Rows If B is the matrix obtained by interchanging any two rows (columns) of an n x n matrix A, then det B = -det A. Theorem: Constant Multiple of a Row If B is the matrix obtained from an n x n matrix A by multiplying a row (column) by a non zero real number k, then det B = k det A. Theorem: Determinant of a Matrix Product If A and B are both n x n matrices, then det AB = det A x det B Theorem: Determinant is unchanged If matrix B is obtained from multiplying entries from row(column) and adding corresponding entries to another row then the det B = det A. Theorem: Determinant of a triangular matrix If matrix A is an n x n triangular matrix (upper or lower). Then det A = a₁1₁a22...ann where a11a22...ann are the entries on the main diagonal of A. Theorem: A Property of Cofactors Suppose A is an n x n matrix. If we get the cofactors of the entries in the kth row. When we obtain the cofactor for the jth column a₁1Ck1 + a2Ck2 +...+ ainCkn = 0, for i #k a1jC1k + a2jC2k +...+ anjCnk = 0, for j #k
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