Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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PLEASE SHOW COMPLETE SOLUTION AND PLS WRITE LEGIBLY. NO SHORTCUTS
![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](https://content.bartleby.com/qna-images/question/dbb55b9f-e380-4b90-858b-1fcdc5cfe339/aec7dccc-5e93-4f11-82b8-6c98643069de/lr0g2ku_processed.png)
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
![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.](https://content.bartleby.com/qna-images/question/dbb55b9f-e380-4b90-858b-1fcdc5cfe339/aec7dccc-5e93-4f11-82b8-6c98643069de/qnqprq_processed.png)
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.
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