I Solve the following problems. Show your detailed solution. 1. Prove that the properties of determinants are true by solving the following: 3 1 -5 01 5 -4 4 3 --5 -1 -4 4 A = 2 3 -5 B1 = 6 -3 -4 4 3 B2 = -3 -6 -6 -6 -6 5 4 -2 -4 --6 k= -2 1 7 -1 1 2 Answer 6. -21 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 [B1] 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 covenient method when solving the determinant of the given matrices
I Solve the following problems. Show your detailed solution. 1. Prove that the properties of determinants are true by solving the following: 3 1 -5 01 5 -4 4 3 --5 -1 -4 4 A = 2 3 -5 B1 = 6 -3 -4 4 3 B2 = -3 -6 -6 -6 -6 5 4 -2 -4 --6 k= -2 1 7 -1 1 2 Answer 6. -21 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 [B1] 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 covenient method when solving the determinant of the given matrices
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Answer #1 subpart d
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbce54958-74b6-4984-b6f0-9f5ded78a01d%2F46452747-36c3-4c8f-8d59-d0bc15ac28e3%2Fppf278_processed.jpeg&w=3840&q=75)
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
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbce54958-74b6-4984-b6f0-9f5ded78a01d%2F46452747-36c3-4c8f-8d59-d0bc15ac28e3%2Ftegu878_processed.jpeg&w=3840&q=75)
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
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