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Consider the two 3 × 4 matrices below
1. Row-reduce each matrix and determine that the reduced row-echelon forms of B and C are identical. From this argue that B and C are row-equivalent.
2. In the proof of Theorem RREFU, we begin by arguing that entries of row-equivalent matrices are related by way of certain scalars and sums. In this example, we would write that entries of B from row i that are in column j are linearly related to the entries of C in column j from all three rows
[B]ij = δi1 [C]1j + δi2 [C]2j + δi3 [C]3j 1 ≤ j ≤ 4
For each 1 ≤ i ≤3 find the corresponding three scalars in this relationship. So your answer will be nine scalars, determined three at a time.
Physics
Contributed by Robert Beezer Statement [44]
1. Let R be the common reduced row-echelon form of B and C. A sequence of row operations converts B to R and a second sequence of row operations converts C to R. If we \reverse" the second sequence’s order, and reverse each individual row operation (see Exercise RREF.T10) then we can begin with B, convert to R with the first sequence, and then convert to C with the reversed sequence. Satisfying Definition REM we can say B and C are row-equivalent matrices.
2. We will work this carefully for the first row of B and just give the solution for the next two rows. For
row 1 of B take i = 1 and we have
[B]1j = δ11 [C]1j + δ12 [C]2j + δ13 [C]3j 1 ≤ j ≤ 4
If we substitute the four values for j we arrive at four linear equations in the three unknowns δ11, δ12, δ13,
We form the augmented matrix of this system and row-reduce to find the solutions,
So the unique solution is δ11 = 2, δ12 = −3, δ13 = −2. Entirely similar work will lead you to
δ21 = −1 δ22 = 1 δ23 = 1
δ31 = −4 δ32 = 8 δ33 = 5
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