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Elementary Matrices, Invertible Matrices and Systems of Equations

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.

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