Hi, I would like to have the answers to the following question. Please write down detailed explanation for me. Algebra(a) Suppose A is a 3 × 3 matrix such that AUsing theorems from the course, find the determinant of A – 21(and explain your answer with reference to the course results).(b) Leta b||A = d e f9 haiwith real numbers a, b, c, d, e, f, g, h, i such that det(A) = 3.aWhat is the determinant of B = d+2a2gGive a justification for your answer.bCe+2b ƒ+2c ?2h 2i

Answered: Image /qna-images/answer/2171a246-0030-4740-ac6f-924d191fc63a.jpg

Answered: A= = de f hi 9 with real numbers a, b,…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter3: Determinants

Section3.CR: Review Exercises

Problem 55CR

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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.
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.

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A= = de f hi 9 with real numbers a, b, c, d, e, f, g, h, i such that det(A) = 3. a b C What is the determinant of B = d+2a e+2b f+2c ?