If A and B are 3 × 3 real matrices such that rank(AB) = 1, then rank(BA) cannot be

To analyze the problem regarding the ranks of the matrices $A$ and $B$, we will break down the problem as follows:

1. Break Down the Problem

We know that:

• $A$ and $B$ are $3 \times 3$ matrices.
• $\text{rank}(AB) = 1$.

We need to determine the possible values of $\text{rank}(BA)$.

2. Relevant Concepts

We can use the following key concepts about ranks of matrices:

• The rank of a product of matrices is less than or equal to the minimum of the ranks of the individual matrices: $\text{rank}(AB) \leq \min(\text{rank}(A), \text{rank}(B))$
• For two matrices $A$ and $B$: $\text{rank}(AB) = \text{rank}(BA)$ when both matrices have full row rank.

3. Analysis and Detail

Given that $\text{rank}(AB) = 1$, we can conclude:

• Since $\text{rank}(AB) \leq \min(\text{rank}(A), \text{rank}(B))$, we have $\text{rank}(A) \geq 1 \quad \text{and} \quad \text{rank}(B) \geq 1.$
• If one of the ranks (either $A$ or $B$) were 1, the other could at most be 2, since the rank of the product is limited by the matrix with the lower rank.

Now, we will analyze what this implies for $\text{rank}(BA)$:

• Generally, if $\text{rank}(A) = 1$ and $\text{rank}(B) \leq 2$, then
  • $\text{rank}(BA)$ could also potentially be 1, 0, or even fall below the rank of $B$.
• However, if $\text{rank}(B) = 1$, then $\text{rank}(BA)$ would still lead to a rank that could be 1 or 0.

Given these considerations, the possible ranks for $BA$:

• The rank of $BA$ can be either 0, 1, or 2 based on the ranks of $A$ and $B$.

4. Verify and Summarize

• The critical insight is that $\text{rank}(BA)$ cannot be 3 since the ranks of $A$ and $B$ are constrained by the fact that $\text{rank}(AB) = 1$.

Final Answer

Therefore, the rank of $BA$ cannot be $3$.