 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2014/digital-systems-linear-algebra-introduction-to-formal-proof/6/b/solution

Use

R_{2} \leftarrow R_{2} + 2 R_{1}

and

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

, followed by

R_{3} \leftarrow R_{3} + R_{2}

1 - 2 3 4 - 4 8 100 \sim 100 44 - 4 12 - 3 \sim U 100440 12 - 1 .

m_{21} = - 2

m_{31} = 3

m_{32} = - 1

L

A = LU, L = 1 - 2 3 01 - 1 001, U = 100440 12 - 1 .

To see the elementary matrix relation directly, let

F_{21} = I + 2 e_{21}

F_{31} = I - 3 e_{31}

and

F_{32} = I + e_{32}

, where

e_{ij}

(i, j)

. Then

U = F_{32} F_{31} F_{21} A

, and reversing and inverting these operations gives

L = F_{21}^{- 1} F_{31}^{- 1} F_{32}^{- 1}

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