EXERCICE 3: Linear maps and determinantFor all aeR, let / R³ R³ be the linear map defined byfa (x, y, z) = (x + (2(1) Write the matrix A, off in the canonical basis.(2) Compute the determinart of Age)y+z, xy +z₁x - y + (4- a)z).(3) Determine the values of a such that fa is injective.(4) Determine the values of a such that (1,1,1) belongs to Im(fa).(5) Fora = 1, determine the kernel and the image of ₁.(6) Construct a linear map g: R² R³ such that/m g = Im fo

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Answered: EXERCICE 3: Linear maps and determinant…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter3: Determinants

Section3.1: The Determinants Of A Matrix

Problem 44E: True or False ? a To find the determinant of a triangular matrix, add the entries on the main...

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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.
Find the determinant of the matrix in Exercise 15 using the method of expansion by cofactors. Use a the second row and b the second column. 15. [321456231]
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]
The Determinant of a scalar multiple of a Matrix In Exercises 7-14, use the fact that |cA|=cn|A| to evaluate the determinant of the nn matrix. A=[5151020]
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
Compute the determinants in Exercises 7-15 using cofactor expansion along any row or column that seems convenient. 14.
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
Finding a Determinant In Exercises 2124, use either elementary row or column operations, or cofactor expansion, to find the determinant by hand. Then use a software program or a graphing utility to verify your answer. |5101101120121031|
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.
Compute the determinants in Exercises 7-15 using cofactor expansion along any row or column that seems convenient. 11.
Proof If A is an idempotent matrix (A2=A), then prove that the determinant of A is either 0 or 1.
True or false? det(A) is defined only for a square matrix A.

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