1 System of Linear EquationsGiven matrix A € Rmxn and column vector bRmx1 describe the set X of all v € R¹x1that solves Au = b. Remember that X = 0 (empty set, meaning there is no v that solvesthe system of equations) iff b span(A.,₁,..., A.), otherwise there exists a solution. If asolution to exists, then the set of all the solutions can be expressed as X = 2o+null(A), wherenull(A) = span (v₁,..., Uk), and U₁,..., U is a maximal set of linearly independent solutionsof the homogeneous equation Av = 0. The dimension k of null(A) equals to n - rank(4),i.e. the number of variables minus the number of independent rows/columns of A.You can use a formula for A-¹, e.g. https://www.cuemath.com/algebra/inverse-of-2x2-matrix/.Don't forget to consider all the cases based on the values of the parameters a, 3 € R.(a)A =b=5A = [51], b= [8]6A =- [8]6A =- [8]164-B 3 --AA =b=2(b)(c)(d)(e)b=b= (f)09A =b =A =[1 2 3]4 5 6b=(h)1 2 3A =b=[]748 aHint: To find examples of vectors v = [x, y, z] that solve the homogeneous equationAv = 0, you can deliberately set some of the variables to be equal 0 or 1:if rank(A) = 2, then the dimension of null(A) is 1, so you only need to find onevector ₁ and then null(A) = span(v₁). Try to put r = 1 and solve the rest of theequation. If the equation did not have a solutions, then you would put y = 1 orz = 1 instead.• if rank(A) = 1, then you can pick values of two of the variables x, y, z. Rememberthat now you need to find two independent vectors to span null(A). For example,put (x, y) = (1,0) and try find z; and then put (x, y) = (0, 1) and try find z.

Answered: Image /qna-images/answer/febb6fa1-15fd-49e3-857a-d2653a88c910.jpg

Answered: b= A = [51], b= [8] 6 A = b= -B -2 1 α…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter6: Matrices And Determinants

Section: Chapter Questions

Problem 66E: Distribution of Cash An ATM at a bank in Qualicum Beach, British Columbia, dispenses $20 and $50...

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CAPSTONE The dimension of the row space of a 35 matrix A is 2. a What is the dimension of the column space of A? b What is the rank of A? c What is the nullity of A? d What is the dimension of the solution space of the homogeneous system Ax=0?
Guided Proof Prove that if A is row-equivalent to B, and B is row-equivalent to C, A is row-equivalent to C. Getting Started: to prove that If A is row-equivalent to C, you have to find elementary matrices E1, E2. Ek such that A=EkE2E1C. i Begin by observing that A is row-equivalent to B and B is row-equivalent to C. ii This means that there exist elementary matrices F1F2Fn and G1G2Gm such that A=FnF2F1B and B=GmG2G1C. iii Combine the matrix equations from step ii.
(a) The matrix I=[1001] is called an _____ matrix. (b) If A is a 22 matrix, then AI= _____ and IA= _____. (c) If A and B are 22 matrices with AB=I , then B is the _____ of A.
Proof Let A be an mn matrix. a Prove that the system of linear equations Ax=b is consistent for all column vectors b if and only if the rank of A is m. b Prove that the homogeneous system of linear equations Ax=0 has only the trivial solution if and only if the columns of A are linearly independent.
Distribution of Cash An ATM at a bank in Qualicum Beach, British Columbia, dispenses $20 and $50 bills. Brodie withdraws $600 from this machine and receives a total of 18 bills. Let x be the number of $20 bills and y the number of $50 bills that he receives. (a) Find a system of two linear equations in x and y that express the information given in the problem. (b) Write your linear system as a matrix equation of the form AX=B . (c) Find A1 , and use it to solve your matrix equation in part (b). How many bills of each type did Brodie receive?
Electricity By using Kirchhoffs Laws, it can be shown that the currents I1, I2 and I3 that pass through the three branches of the circuit in the figure satisfy the given linear system. Solve the system to find I1, I2 and I3. {I1+I2I3=016I18I2=48I2+4I3=5
Guided Proof Prove that the inverse of a symmetric non-singular matrix is symmetric. Getting Started: To prove that the inverse of A is symmetric, you need to show that A-1T=A-1. i Let A be symmetric, nonsingular matrix. ii This means that AT=A and A-1 exists. iii Use the properties of the transpose to show that A-1T is equal to A-1.
Fill in the blanks. If A is an invertible matrix, then the system of linear equations represented by AX=B has a unique solution given by X=.
(a) How do Gaussian elimination and Gauss-Jordan elimination differ? (b) Use Gauss-Jordan elimination to solve the linear system in part 3(d).

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b= A = [51], b= [8] 6 A = b= -B -2 1 α B A = -1/2 1 6 3 -2 -1 A = b=