1. Assume you are given a matrix in reduced row echelon form, with r nonzero row
ID: 3122297 • Letter: 1
Question
1. Assume you are given a matrix in reduced row echelon form, with r nonzero rows and c nonzero columns. Which of these must be true? r<=c,r=c,r>=c, Why?
2. Suppose that A is the augmented matrix of a system of linear equations. Prove that if the last (augmenting) column of A is a linear combination of the other columns then the system of equations has at least one solution.
3. True or false: If A does not reduce to B and B does not reduce to C, then A does not reduce to C. (Give a proof or counterexample.) As always, be sure to include justifications for your answers.
Explanation / Answer
1. A matrix is said to be in row echlon form, if the following conditions are satisfied.
a) All rows with at least one non zero element should be above the row with all zeros.
b) The first nonzero number of the left side of the non zero row should be to the right side of the leading coefficient of the row above it.
Inorder to satisfy these conditions, number of rows should be less than or equal to the number of columns.
That is r < = c or r= c.
If r> c, it is impossible to satisfy the second condition.
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