Practice: Elimination and A = LU
Three worked examples, then five problems. Do the elimination by hand first. The lab has lu(A) and rref(A) to check your answers.
Worked examples
Example 1 · A 2 by 2 system
Solve 2x + 3y = 8 and 4x + 11y = 26.
The pivots are 2 and 5. Their product, 10, is det A.
Example 2 · A 3 by 3 system
Solve x + y + z = 6, 2x + 3y + z = 11, x + 3y + 4z = 19.
U = 11101−1005, pivots 1, 1, 5.
Example 3 · Factor once, solve with L and U
Factor A = 2168 and use it to solve Ax = (3, 19).
Problems
Solve x − y = 1 and 3x + 2y = 13 by elimination. Name the pivots.
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ℓ = 3. Equation 2 − 3·(equation 1): 5y = 10, so y = 2 and x = 3. Check: 9 + 4 = 13 ✓
Pivots: 1 and 5.
For A = 4265, find the multiplier ℓ₂₁, the second pivot and det A.
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ℓ₂₁ = 6 ÷ 4 = 1.5. Second pivot: 5 − 1.5·2 = 2.
det A = 4·2 = 8, the product of the pivots. Check: 4·5 − 2·6 = 8 ✓
For which number a does 1a28 have no second pivot? Can a row exchange fix it?
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The second pivot is 8 − 2a. It is 0 when a = 4.
No row exchange helps: the only row below is already used. The matrix is singular (column 2 = 4·column 1).
Write the elimination matrix E₂₁ for A = 210641023 and compute E₂₁A.
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ℓ₂₁ = 6 ÷ 2 = 3, so E₂₁ = 100−310001.
E₂₁A = 210011023. Only row 2 changed: (6, 4, 1) − 3·(2, 1, 0) = (0, 1, 1).
Find L and U for A = 12310.
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ℓ = 3, so U = 1204 and L = 1031.
Check: LU = 1236 + 4 = A ✓
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