Let A, B and C be three matrices all of whose entries are 1s. A is a 5x5 matrix, B is a 3x2 matrix and C is a 666x667 matrix. Which of these matrices has an inverse?

Question

BlackTom AI · Accepted Answer

We start by identifying the properties that determine whether a matrix has an inverse.

Option a. None
- The statement claims that none of A, B, or C has an inverse. To evaluate this, we examine each matrix:
- A is 5x5 with every entry equal to 1. Such a matrix is the all-ones matrix J_5. The rank of J_n is 1 for n > 1, since every row is identical and can be written as a multiple of any other row. A square matrix with rank less than its size is singular and thus not invertible. Therefore A does not have an inverse.
- B is 3x2, which is not a square matrix. A matrix must be square to have an inverse in the standard linear-algebra sense; non-square matrices do not have inverses.
- C is 666x667, also not square. Like B, a non-square matrix cannot have an inverse.
From these points, A is not invertible, and B and C cannot have inverses because they are non-square. Thus the claim in option a can be evaluated as consistent with the above facts.

Option b. C
- C is 666x667, a non-square matrix. Non-square matrices do not possess inverses in the usual sense, so this option cannot be correct because it asserts an inverse exists for a non-square matrix.

Option c. B
- B is 3x2, which is also non-square. As with C, the inverse in the standard sense does not exist for non-square matrices. Therefore this option is invalid.

Option d. A
- A is 5x5 and square, but as discussed, the all-ones 5x5 matrix has rank 1, not full rank, making it singular and non-invertible. Hence this option cannot be correct.

Across all options, only square matrices can potentially have inverses, and among the given matrices, the only square one (A) is not invertible due to singularity. The non-square matrices (B and C) cannot have inverses by definition. Consequently, the overall assessment aligns with the claim that none of A, B, or C has an inverse. The reasoning hinges on the rank deficiency of the all-ones square matrix and the non-square nature of the other two matrices.

Let A, B and C be three matrices all of whose entries are 1s. A is a 5x5 matrix, B is a 3x2 matrix and C is a 666x667 matrix. Which of these matrices has an inverse?单项选择题

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