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.
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.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Determinants
Section3.1: The Determinants Of A Matrix
Problem 70E: The determinant of a 22 matrix involves two products. The determinant of a 33 matrix involves six...
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![Prove that the properties of determinants are true by solving the following:
-1
1
3
1
-5
-5
-1
-4
4
-4
4
3
-7
-4
4
3
A =
2
-5
B1
B2
-3
-6
-6
3
-6
-6
5
6
-3
-2]
6-
4
-4 -3
1
4
-4
-3.
-2 -3
-2
-4
-6
k = -2
C =
1
7
-1
D :
1
2
3
4
-1
3
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 [D] 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 convenient method when solving the determinant of the given matrices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdbb55b9f-e380-4b90-858b-1fcdc5cfe339%2F38f3fd35-b328-4929-bc03-d76a8a7a4601%2Fsld0rr_processed.png&w=3840&q=75)
Transcribed Image Text:Prove that the properties of determinants are true by solving the following:
-1
1
3
1
-5
-5
-1
-4
4
-4
4
3
-7
-4
4
3
A =
2
-5
B1
B2
-3
-6
-6
3
-6
-6
5
6
-3
-2]
6-
4
-4 -3
1
4
-4
-3.
-2 -3
-2
-4
-6
k = -2
C =
1
7
-1
D :
1
2
3
4
-1
3
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 [D] 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 convenient method when solving the determinant of the given matrices.
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdbb55b9f-e380-4b90-858b-1fcdc5cfe339%2F38f3fd35-b328-4929-bc03-d76a8a7a4601%2Fcgrv39m_processed.png&w=3840&q=75)
Transcribed Image Text: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
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