 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/digital-systems-linear-algebra-introduction-to-formal-proof/3/c/solution

Use Gaussian elimination without row exchanges. First apply

R_{2} \leftarrow R_{2} - 3 R_{1}

and

R_{3} \leftarrow R_{3} - 2 R_{1}

; then apply

R_{3} \leftarrow R_{3} - R_{2}

13228641413 ⟶ 100222425 ⟶ U = 100220423 .

m_{21} = 3

m_{31} = 2

and

m_{32} = 1

L = 132011001, U = 100220423, A = LU .

To see why these multipliers reconstruct the rows, row 2 of

LU

is three times row 1 of

U

plus row 2 of

U

(3, 8, 14)

. Row 3 of

LU

is twice row 1 plus row 2 plus row 3 of

U

(2, 6, 13)

. This directly verifies the LU factorization.

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