Hi, I would like to have the solutions to the questions below. Can you write down detailed explanation for me, please? Algebra(a) Suppose A is a 3 × 3 matrix such that AHUsing theorems from the course, find the determinant of A - 21(and explain your answer with reference to the course results).(b) LetA =||a b cde fLg h iwith real numbers a, b, c, d, e, f, g, h, i such that det(A) = 3.What is the determinant of BN28Give a justification for your answer.abс4 + 24d+2a e+2b f+ 2c ?2g2h2i

Answered: Image /qna-images/answer/d3cc6d69-0dcd-4158-acec-ba0fd34db59f.jpg

Answered: Algebra (a) Suppose A is a 3 × 3 matrix…

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
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.
Examining Cramer's Rule, explain why there is no unique solution to the system when the determinant of your matrix is O. For simplicity, use a 22 .
The Determinant of a Matrix In Exercises 1-12, find the determinant of the matrix. [76123]
Collinear Points and Determinants a If three points lie on a line, what is the area of the triangle that they determine? Use the answer to this question, together with the determinant formula for the area of a triangle, to explain why the points (a1,b1), (a2,b2) and (a3,b3) are collinear if and only if |a1b11a2b21a3b31|=0 b Use a determinant to check whether each set of points is collinear. Graph them to verify your answer. i (6,4), (2,10), (6,13) ii (5,10), (2,6), (15,2)
The Determinant of the Inverse of a Matrix In Exercises 29-32, find |A-1|. Begin by finding A-1, and then evaluate its determinant. Verify your result by finding |A|and then applying the formula from Theorem 3.8, |A-1|=1|A|. A=[112248110]

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