Compute the determinant of the matrix A, below, by using row operations to transform A to an upper-triangular matrix B. Then express thedeterminant of A as a multiple k of the determinant of B and use this to compute the determinant of A.46 -3 -8-8 -14 19 34A =0 5-111 -70 0 0B = 0 0 00 0 0det(A) = k-det(B)= 0 0= 0

Answered: Compute the determinant of the matrix…

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.CR: Chapter Review

Problem 11CC

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True or False ? a The determinant of a 22 matrix A is a21a12a11a22. b The determinant of a matrix of order 1 is the entry of the matrix. c The ij-cofactor of a square matrix A is the matrix obtained by deleting the ith row and jth column of A
Find the determinant of the matrix in Exercise 16 using the method of expansion by cofactors. Use a the third row and b the first column. 16. [342631478]
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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
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.

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Compute the determinant of the matrix A, below, by using row operations to transform A to an upper-triangular matrix B. Then express the determinant of A as a multiple k of the determinant of B and use this to compute the determinant of A. 4 6 -3 -8 -8 -14 19 34 A = 5 -1 1 |1 -7 0 0 0