Given
𝐵
=
[
−
1

0

1

0

1

−
1
]
, please find Cov(AX, BX).

Question

Given 
𝐵
=
[
−
1
	
0
	
1

0
	
1
	
−
1
]
, please find Cov(AX, BX).

BlackTom AI · Accepted Answer

The question asks for Cov(AX, BX) given a matrix B = [ [-1, 0, 1], [0, 1, -1] ]. To reason about this, recall a standard identity: Cov(AX, BX) = A Cov(X) B^T, assuming X is a random vector with Cov(X) well-defined and A, B are conformable matrices.

Key clarifications needed to evaluate the options:
- We need the matrix Cov(X) (often denoted Σ or Cov(X)) to compute Cov(AX, BX) via the formula A Σ B^T.
- We also need the matrix A to apply the left multiplication and the shapes to match (A must have rows corresponding to the dimension of AX and columns matching the number of rows of X, similarly for B with BX).
- In the provided data, A is not given, nor is Cov(X) specified. Without these, Cov(AX, BX) cannot be numerically determined. Any concrete numeric option would implicitly assume a specific A and Σ, which aren’t provided here.

Now, let’s examine the options in light of this:
Option 1: [
−4
−3

−2
1

−8
7
] 
This appears to be a 3-by-2 block-like arrangement. As a covariance matrix, dimensions must be square matrices (Cov returns a square matrix with size equal to the dimension of the random vector, here the dimension of AX or BX). Without knowing the target dimension of AX and BX, this option’s shape is hard to justify as a covariance matrix. Moreover, a covariance matrix must be symmetric and positive semidefinite; the given arrangement does not obviously convey symmetry, so this raises red flags about feasibility.

Option 2: None of these options are correct.
This is a meta-answer asserting that none of the numerical matrices shown can be correct. If indeed Cov(AX, BX) depended on unknown A and Cov(X), it would be impossible to verify any concrete numeric matrix as correct without those inputs. Claiming that none are correct could be valid if all presented matrices fail basic covariance-matrix properties or fail to match any possible A Σ B^T for any reasonable shapes, but we don’t have enough data to verify that claim definitively.

Option 3: [
−4
−3

−8
7
]
This looks like a 2-by-2 arrangement, which is a plausible shape for a covariance matrix if AX and BX are each 1-dimensional (i.e., a 1x1 variance with a 2x2 cross-covariance structure would be unusual). Still, any covariance matrix must be symmetric; here the arrangement does not immediately display symmetry unless the formatting is deceptive. Without knowing A and Cov(X), we cannot confirm symmetry or definiteness, so this option cannot be validated in isolation.

Option 4: [
4
−3

−2
1
]
Again, a 2-by-2-looking arrangement with entries that would need to satisfy symmetry (the (1,2) and (2,1) entries should be equal in a covariance matrix). The lack of symmetry in the textual formatting here makes it unreliable as a covariance candidate without further context. As with the others, the feasibility depends on A and Cov(X).

Option 5: [
−4
−3

−8
7

−2
1
]
This appears to be a larger matrix with more entries; as a covariance matrix, it must be square and symmetric with nonnegative eigenvalues. The presence of seemingly inconsistent block structure and potential asymmetry again raises questions about validity without knowing the A and Cov(X) used to generate it.

Option 6: [
8
−7

−2
1
]
A final 2-by-2 attempt; again, symmetry is key for a covariance matrix, and the listed entries do not obviously ensure symmetry. Without A and Cov(X), we cannot confirm whether this could be Cov(AX, BX).

Overall assessment:
- The core obstacle is missing information: A and Cov(X). Cov(AX, BX) = A Cov(X) B^T, so every candidate answer would need to be expressible as A Σ B^T for some A and Σ that align with the given B. Since A and Σ are not provided, we cannot verify any numeric candidate as correct.
- Additionally, covariance matrices must be square, symmetric, and positive semidefinite. Several presented formats do not obviously satisfy these properties, or their shapes are unclear from the textual formatting, making them dubious as bona fide covariance results.
- The provided “answer” field in the dataset shows a specific string: [−4−3−87], which resembles a concatenated vector or a misformatted matrix entry sequence. Without proper dimensional context, this string cannot be confidently interpreted as a valid covariance result.

In summary, due to missing information (A and Cov(X)) and the ambiguous formatting of the answer options, it is not possible to determine which option (if any) is correct for Cov(AX, BX) from the data given. If you can supply the matrices A and Cov(X) (or the distribution of X), I can compute Cov(AX, BX) precisely and map it to the correct option among the choices. If Cov(X) is the identity and A is known, we could also compute AB^T and compare to the options more concretely.

25S-STATS-102B-LEC-3 Final Exam- Requires Respondus LockDown Browser

Given 𝐵 = [ − 1 0 1 0 1 − 1 ] , please find Cov(AX, BX).

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