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1. Prove that the properties of determinants are true by solving the following: [3 1 -5 o -5 [-1 -4 4 A =2 3 -5 B1 =5 -4 4 3 -3 -6 -6 5 -4 4. 3 B2 = -6 -6 6 -3 -2 4 -4 -3. 4 -4 -3. -4 swer D =1 2 3 6 -21. k=-2 a. Using Matrix [A] prove that the Property 1 and 2 of determinant are true. b. Using Matrix [C] and 'k prove that Property 3 of determinant is true. C Using Matrix Dj prove that Property 4 of determinant is true.

1. Prove that the properties of determinants are true by solving the following: [3 1 -5 o -5 [-1 -4 4 A =2 3 -5 B1 =5 -4 4 3 -3 -6 -6 5 -4 4. 3 B2 = -6 -6 6 -3 -2 4 -4 -3. 4 -4 -3. -4 swer D =1 2 3 6 -21. k=-2 a. Using Matrix [A] prove that the Property 1 and 2 of determinant are true. b. Using Matrix [C] and 'k prove that Property 3 of determinant is true. C Using Matrix Dj prove that Property 4 of determinant is true.

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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Answer #1 in the second pic (where I wrote "answer") rush pls thanks
PROPERTIES OF DETERMINANTS 1. Determinant of a Transpose The determinant of a transpose A" of A is equal to the determinart 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. E:9-E ja21 az2 a1 a12 *** azn ain ain a21 Jani an2 annl Jant anz 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: [an a12 a a12 ain az1 a2 A = A' = ... ajz ba ba b Then: det(A) + det(A') = au + bu aa + b2 an + bin ain lani anz Lant an2 ann ant an2 ann
Transcribed Image Text:PROPERTIES OF DETERMINANTS 1. Determinant of a Transpose The determinant of a transpose A" of A is equal to the determinart 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. E:9-E ja21 az2 a1 a12 *** azn ain ain a21 Jani an2 annl Jant anz 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: [an a12 a a12 ain az1 a2 A = A' = ... ajz ba ba b Then: det(A) + det(A') = au + bu aa + b2 an + bin ain lani anz Lant an2 ann ant an2 ann
I Solve the following problems. Show your detailed solution. 1. Prove that the properties of determinants are true by solving the following [-1 -4 4 3 1 -5 -5 A =2 3 -5 B1 =5 -4 4 3 -6 5 -4 -31 -3 6-1 4 B2 =-7 -4 4 -6 -6 16 -3 -21 -6 3 4 -4 -3) -3 [-2 -4 D=1 2 3. 3 6 -21l -6 C=1 7 3 4 k= -2 Answer a Using Matrix [A] prove that the Property 1 and 2 of determinant are true. b. Using Matrix [C] and k prove that Property 3 of determinant is true. C Using Matrix [Dj prove that Property 4 of determinant is true. d. Using Matrix [81] and (B2) prove that by solving the determinant of these matrices separately and getting its sum will give the same answer with the use of Property 5. NOTE Use your convenient method when solving the determinant of the given matrices
Transcribed Image Text:I Solve the following problems. Show your detailed solution. 1. Prove that the properties of determinants are true by solving the following [-1 -4 4 3 1 -5 -5 A =2 3 -5 B1 =5 -4 4 3 -6 5 -4 -31 -3 6-1 4 B2 =-7 -4 4 -6 -6 16 -3 -21 -6 3 4 -4 -3) -3 [-2 -4 D=1 2 3. 3 6 -21l -6 C=1 7 3 4 k= -2 Answer a Using Matrix [A] prove that the Property 1 and 2 of determinant are true. b. Using Matrix [C] and k prove that Property 3 of determinant is true. C Using Matrix [Dj prove that Property 4 of determinant is true. d. Using Matrix [81] and (B2) prove that by solving the determinant of these matrices separately and getting its sum will give the same answer with the use of Property 5. NOTE Use your convenient method when solving the determinant of the given matrices
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