Under LU factorisation of

[
8

2

2

4

4

2

0

1

3
]

write down the value of
𝐿
21
(correct to 1 decimal place)

Question

Under LU factorisation of

[
8
	
2
	
2

4
	
4
	
2

0
	
1
	
3
]

write down the value of 
𝐿
21
 (correct to 1 decimal place)

BlackTom AI · Accepted Answer

LU factorization writes A = L U with L a unit lower-triangular matrix (1s on the diagonal). The entry L21 is the multiplier used to eliminate the (2,1) entry of A using the first pivot a11.

1) Relevant concepts
- Doolittle LU form: L has 1s on the diagonal and multipliers lij = aij / pivot used to eliminate below-pivot entries.
- The first multiplier L21 = a21 / a11.

2) Step-by-step reasoning and row operations
Start with A =
[8  2  2
 4  4  2
 0  1  3].

The first pivot is a11 = 8. To eliminate a21 = 4, compute the multiplier:
L21 = a21 / a11 = 4 / 8.

3) Intermediate calculation
4 / 8 = 0.5.

Apply the row operation R2 <- R2 - L21 * R1 to produce the first elimination step (this shows the multiplier in action):
R1 = [8, 2, 2]
R2 = [4, 4, 2] − 0.5*[8, 2, 2] = [4 − 4, 4 − 1, 2 − 1] = [0, 3, 1].

(Continuing to form full U would give R3 unchanged initially and further multipliers for the (3,2) elimination, but those are not needed to find L21.)

4) Why this answer is correct
By definition in the chosen LU convention (unit lower triangular L), L21 is exactly the ratio of the (2,1) element to the (1,1) pivot. That ratio evaluates to 0.5, so L21 = 0.5. Rounded to one decimal place, this remains 0.5.

5) Other possibilities
No alternative option applies here; only if a different factorization convention or row pivoting were used could entries differ. Under the standard unit-L (Doolittle) LU without pivoting, the computed value is uniquely 0.5.

6) Conclusion
The computed value L21 = 4/8 = 0.5 (to one decimal place). This matches the provided correct answer 0.5.

Under LU factorisation of [ 8 2 2 4 4 2 0 1 3 ] write down the value of 𝐿 21 (correct to 1 decimal place)数值题

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