Elimination matrices: each row step is a matrix
Slides 2 to 4 did elimination by hand, one row step at a time. Now we write each step as a matrix, so a computer can store the steps and reuse them. Think of an instruction card: “take 2 times row 1 away from row 2, leave the other rows alone”. An elimination matrix is that card written in numbers. It starts from the identity matrix I (1s on the diagonal, 0s elsewhere), the card that says “keep every row as it is”. Put −ℓ in row 2, column 1, and you get E₂₁. Here E₂₁A means one simple thing: each row of E₂₁ says how to combine the rows of A. Slide 6 shows how to multiply any two matrices.
Lesson 2 · slide 5Elimination matrices- An elimination matrix E is the identity with one extra entry, −ℓ, below the diagonal (if ℓ is negative, this entry is positive).
- Multiplying EA performs that row step on A.
- A row of EA is a combination of the rows of A.
- To undo E, change −ℓ to +ℓ. This is the key to A = LU on slide 7.
Why write a step as a matrix?
As matrices, the whole list of steps becomes one equation that a computer can store and reuse.
An instruction card in numbers
E₂₁ is the card "subtract 2 times row 1 from row 2, leave the rest alone", written as numbers. Multiplying by it follows the card.
Why it works: rows combine rows
This is a new rule, the mirror image of lesson 1's column rule. There, Ax combined the columns of A. Here, row i of EA is a combination of the rows of A, and row i of E gives the weights. (Slide 6 shows this matches the usual row-times-column rule.)
- Row 1 of E₂₁ is 100: keep row 1.
- Row 2 is −210: row 2 minus 2 times row 1.
- Row 3 is 001: keep row 3.
A row times a matrix uses the row's numbers as weights on the matrix's rows. Row 2, with A from slide 3:
That is the new row 2 from slide 3. The same rule explains why I changes nothing: row i of I has a single 1, in spot i, so it picks out row i of A. So IA = A.
The whole process as one product
Read it right to left: E₂₁ is closest to A, so it acts first. Then E₃₁ acts on that result, then E₃₂. Each step uses only the rows-combine-rows rule; we never need to multiply the E's with each other here.
Undoing a step
To undo E₂₁, add 2 times row 1 back. We call that matrix E₂₁⁻¹. For now, ⁻¹ just means “the undo step”; lesson 3 covers inverses. It is the identity with +2 in the same spot. Every elimination matrix is undone by flipping the sign of its extra entry.
Next: So far E·A only needed the rows-combine-rows rule. Slide 6 shows how to multiply any two matrices. Then slide 7 uses the undo steps to factor A = LU.
Examples
Example 1 · A 2 by 2 step
A = 3164. The multiplier is ℓ = 6 ÷ 3 = 2.
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Example 2 · The third step, E₃₂
After the first two steps, B = 2110−8−2083. Clear the 8 with ℓ₃₂ = 8 ÷ (−8) = −1.
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Example 3 · Undoing a step
E₂₁ = 100−210001 subtracts 2·row 1 from row 2.
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Do E₂₁, then E₂₁⁻¹, and A is back where it started. Only the sign of the multiplier changes.
Example 4
Write E₂₁ for A = 2451 and compute E₂₁A.
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ℓ = 5 ÷ 2 = 2.5, so E₂₁ = 10−2.51 and E₂₁A = 240−9. Row 2 of E₂₁A: −2.5·24 + 1·51 = −5−10 + 51 = 0−9.
Example 5
Use the rows-combine-rows rule to check that 10−31 times 1031 gives I.
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Row 1 of the left matrix, 10, keeps row 1 of the right matrix: 10. Row 2, −31, says −3·(row 1) + 1·(row 2) = −3·10 + 31 = 01. The product is 1001.
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