Part i) Write down a system of three equations in three variables, whose augmented matrix [Ab] would then be 3 × 4. Find the determinant of the matrix A whose entries are the coefficients of the system. Ensure the determinant is non-zero and find the inverse of the matrix and also calculate A-¹b (you may use matlab or an online calculator for this part). Describe how A-¹b relates to the system [A]b] in a few words.
Part i) Write down a system of three equations in three variables, whose augmented matrix [Ab] would then be 3 × 4. Find the determinant of the matrix A whose entries are the coefficients of the system. Ensure the determinant is non-zero and find the inverse of the matrix and also calculate A-¹b (you may use matlab or an online calculator for this part). Describe how A-¹b relates to the system [A]b] in a few words.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.1: Matrix Operations
Problem 20EQ: Referring to Exercise 19, suppose that the unit cost of distributing the products to stores is the...
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I have attached an image with a detailed explanation
Matrix A needs to be relatively random looking. So, in
particular it should have:
![• not too many l's or 0's,
• not be in REF or RREF,
• not be a scalar multiple of the identity matrix, or have too many columns which are
scalar multiples of e; the standard basis vectors, and additionally
• should not have a very ugly answer to (Part i).
Part i) Write down a system of three equations in three variables, whose augmented matrix
[A|b] would then be 3 x 4. Find the determinant of the matrix A whose entries are the
coefficients of the system. Ensure the determinant is non-zero and find the inverse of the
matrix and also calculate A-lb (you may use matlab or an online calculator for this part).
Describe how A-lb relates to the system [A|b] in a few words.
Part ii) For the matrix A you wrote dOwn in part (i), row reduce the matrix to rref. What is
the rref of A? Following the row operations you made to reduce A to rref, state the determinant
of each elementary matrix. That is, if I is the rref of A and A ~
E1A
E2E1A ^
..
En .. E2E1A = I are the matrices you got when you row reduced A to I, then calculate or
otherwise find the determinant det A, det E1, det E2, ..., det En, and det I.
Part iii) Compute the determinant of each of the intermediate matrices E¸A, E2E¸A, ...,
En · E2E1A. Compare your result to what you did in part (ii). Try to state the relationship
...
between A and the intermediate row equivalent matrices, in terms of the determinants of the
E¡'s. Formulate a relationship between det(A) and the product of the det(E;)'s.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6e7b8c6-3dfe-46a0-8de7-d0d865f1baff%2F390179d4-af75-4e16-932a-af63d4cd879a%2Ffsfxeg6_processed.png&w=3840&q=75)
Transcribed Image Text:• not too many l's or 0's,
• not be in REF or RREF,
• not be a scalar multiple of the identity matrix, or have too many columns which are
scalar multiples of e; the standard basis vectors, and additionally
• should not have a very ugly answer to (Part i).
Part i) Write down a system of three equations in three variables, whose augmented matrix
[A|b] would then be 3 x 4. Find the determinant of the matrix A whose entries are the
coefficients of the system. Ensure the determinant is non-zero and find the inverse of the
matrix and also calculate A-lb (you may use matlab or an online calculator for this part).
Describe how A-lb relates to the system [A|b] in a few words.
Part ii) For the matrix A you wrote dOwn in part (i), row reduce the matrix to rref. What is
the rref of A? Following the row operations you made to reduce A to rref, state the determinant
of each elementary matrix. That is, if I is the rref of A and A ~
E1A
E2E1A ^
..
En .. E2E1A = I are the matrices you got when you row reduced A to I, then calculate or
otherwise find the determinant det A, det E1, det E2, ..., det En, and det I.
Part iii) Compute the determinant of each of the intermediate matrices E¸A, E2E¸A, ...,
En · E2E1A. Compare your result to what you did in part (ii). Try to state the relationship
...
between A and the intermediate row equivalent matrices, in terms of the determinants of the
E¡'s. Formulate a relationship between det(A) and the product of the det(E;)'s.
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