# Elementary Matrices, Invertible Matrices and Systems of Equations

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1. Consider the matrices

A = 3 4 1 B = 8 1 5 C = 3 4 1

2 -7 -1 2 -7 -1 2 -7 -1

8 1 5 3 4 1 2 -7 3

Is it possible to find an elementary matrix E such that EB = C? Justify your answer in a proof.

2. Write the matrix 3 -2 as a product of elementary matrices.

3 -1

3. Show that if A = 1 0 0 is an elementary matrix, then at least one entry in the

0 1 0 third row must be a zero. Do answer in a proof.

a b c

4. Prove that if A is an invertible matrix and B is row equivalent to A, then B is also invertible. Show answer in a proof.

5. Find the conditions that the b's must satisfy for the system to be consistent.

6x1 -4x2 = b1

6x1 -4x2 = b2

6. Consider the matrices A = 2 1 2 and x = x1

2 2 -2 x2

3 1 1 x3

a) Show that the equation Ax = x can be rewritten as (A - I)x = 0 and use this result to solve Ax = x for x.

b) Solve Ax = 4x

See attached file for full problem description.

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Elementary Matrices, Invertible Matrices and Systems of Equations are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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