Which of the following matrices is not a correct pairwise comparison matrix?

A A1 A2 A3
A1 1 4 7
A2 0.25 1 5
A3 0.1429 0.2 1

B B1 B2 B3
B1 1 0.5 6
B2 2 1 5
B3 0.1667 0.2 1

C C1 C2 C3
C1 1 0.333 0.2
C2 3 1 0.25
C3 5 4 1

D D1 D2 D3
D1 1 5 2
D2 0.4 1 3
D3 0.5 0.3333 1

Question

Which of the following matrices is not a correct pairwise comparison matrix?

A	A1	A2	A3
A1	1	4	7
A2	0.25	1	5
A3	0.1429	0.2	1

B	B1	B2	B3
B1	1	0.5	6
B2	2	1	5
B3	0.1667	0.2	1

C	C1	C2	C3
C1	1	0.333	0.2
C2	3	1	0.25
C3	5	4	1

D	D1	D2	D3
D1	1	5	2
D2	0.4	1	3
D3	0.5	0.3333	1

BlackTom AI · Accepted Answer

We are given four candidate matrices A, B, C, and D, each presented as a pairwise comparison matrix. The key criterion for a valid pairwise comparison matrix in AHP-like frameworks is that the matrix is reciprocal: for every off-diagonal element a_ij, the symmetric element a_ji must satisfy a_ji = 1 / a_ij. Each diagonal element must be 1. Now we examine each option in turn.

Option A:
- Off-diagonal checks show a12 = 4 and a21 = 0.25. Since 1/4 = 0.25, the pair (a12, a21) is reciprocal. Next, a13 = 7 and a31 ≈ 0.1429. Indeed, 1/7 ≈ 0.142857, so a31 ≈ 1/a13 holds. Finally, a23 = 5 and a32 = 0.2, and 1/5 = 0.2 exactly. All off-diagonal pairs satisfy reciprocity. Therefore, A is a valid pairwise comparison matrix.

Option B:
- Check a12 = 0.5 and a21 = 2. Since 1/2 = 0.5, we would expect a21 = 2 if reciprocity holds, which it does here. Next, a13 = 6 and a31 ≈ 0.1667. Since 1/6 ≈ 0.1667, reciprocity holds. Finally, a23 = 5 and a32 = 0.2, with 1/5 = 0.2. All reciprocal pairs are consistent, so B is a valid pairwise comparison matrix.

Option C:
- Evaluate a12 = 0.333 and a21 = 3. Since 1/3 ≈ 0.333, a21 = 3 is the reciprocal of a12. Next, a13 = 0.2 and a31 = 5; 1/5 = 0.2, so reciprocity holds there as well. For the remaining pair, a23 = 0.25 and a32 = 4; 1/4 = 0.25, which matches. All off-diagonal entries satisfy the reciprocal condition, so C is also a valid pairwise comparison matrix.

Option D:
- Look at a12 = 5 and a21 = 0.4. If reciprocity held, we would need a21 = 1/5 = 0.2, but the given value is 0.4, which is not the reciprocal of 5. This breaks the essential reciprocity property of a pairwise comparison matrix. Therefore, D is not a correct pairwise comparison matrix.

Summarizing the individual checks:
- A passes reciprocity checks for all off-diagonal pairs.
- B passes reciprocity checks for all off-diagonal pairs.
- C passes reciprocity checks for all off-diagonal pairs.
- D fails reciprocity for the pair (D1, D2) since a12 ≠ 1 / a21.

Thus, the only option that fails the fundamental requirement is D, making it the incorrect (not a correct) pairwise comparison matrix among the given choices.

Which of the following matrices is not a correct pairwise comparison matrix? A A1 A2 A3 A1 1 4 7 A2 0.25 1 5 A3 0.1429 0.2 1 B B1 B2 B3 B1 1 0.5 6 B2 2 1 5 B3 0.1667 0.2 1 C C1 C2 C3 C1 1 0.333 0.2 C2 3 1 0.25 C3 5 4 1 D D1 D2 D3 D1 1 5 2 D2 0.4 1 3 D3 0.5 0.3333 1单项选择题

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