21-28 - Finding Determinants Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. 2 1 2 5 21. 0 -2 4 22. 2 3 2 -3 3 5 30 20 -2 23. -10 -20 24. 2 4 0 40 10 3 7 -1 25. 2 0 26. 2 6 4 2 2 3 1 3 3 2 1 -4 4 27. 28. -1 6. 1 6 4 1 2. m2 O6

21-28 - Finding Determinants Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. 2 1 2 5 21. 0 -2 4 22. 2 3 2 -3 3 5 30 20 -2 23. -10 -20 24. 2 4 0 40 10 3 7 -1 25. 2 0 26. 2 6 4 2 2 3 1 3 3 2 1 -4 4 27. 28. -1 6. 1 6 4 1 2. m2 O6

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter11: Matrices And Determinants

Section11.CT: Chapter Test

Problem 18CT: TEST Only one of the following matrix has an inverse. Find the determinant of each matrix, and use...

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Question

21-28 - Finding Determinants Find the determinant of the
matrix. Determine whether the matrix has an inverse, but don't
calculate the inverse.
2
1
2 5
21. 0
-2
4
22.
2
3 2
-3
3 5
30
20
-2
23.
-10 -20
24.
2
4 0
40
10
3
7
-1
25. 2 0
26.
2
6 4
2 2
3
1 3
3
2
1
-4
4
27.
28.
-1
6.
1 6
4
1
2.
m2 O6

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TEST Only one of the following matrix has an inverse. Find the determinant of each matrix, and use the determinant to identify the one that has an inverse. Then find the inverse. A=[141020101] B=[140020301]
True or False ? a To find the determinant of a triangular matrix, add the entries on the main diagonal. b To find the determinant of a matrix, expand by cofactors in any row or column. c When expanding by cofactors, you need not evaluate the cofactors of zero entries.
Guided Proof Prove Property 2 of Theorem 3.3: When B is obtained from A by adding a multiple of a row of A to another row of A, detB = detA. Getting Started: To prove that the determinant of B is equal to the determinant of A, you need to show that their respective cofactor expansions are equal. i Begin by letting B be the matrix obtained by adding c times the jth row of A to the ith row of A. ii Find the determinant of B by expanding in this ith row. iii Distribute and then group the terms containing a coefficient of c and those not containing a coefficient of c. iv Show that the sum of the terms not containing a coefficient of c is the determinant of A, and the sum of the terms containing a coefficient of c is equal to 0.
Guided Proof Prove that the determinant of an invertible matrix A is equal to 1 when all of the entries of A and A1 is integers. Getting Started: Denote det(A) as x and det(A1) as y. Note that x and y are real numbers. To prove that det(A) is equal to 1, you must show that both x and y are integers such that their product xy is equal to 1. (i) Use the property for the determinant of a matrix product to show that xy=1. (ii) Use the definition of a determinant and the fact that the entries of A and A1 are integers to show that both x=det(A) and y=det(A1) are integers. (iii) Conclude that x=det(A) must be either 1 or 1 because these are the only integer solutions to the equation xy=1
Find the determinant of the matrix. [4032013501511103]