Given two $n \times n$ invertible square matrices $A$ and $B$ such that $(A-B)(A+B) = A^2-B^2$. Which statement is correct

linear algebramatrices

Given two $n \times n$ invertbible matrices $A$ and $B$ such that $(A − B)(A + B) = A^2 − B^2,$ which of the following statements is correct?

  1. $B = ABA^{-1}B$
  2. None of the given options
  3. $B = A^{-1}BA$
  4. $B = BABA^{-1}$
  5. $A$ is the inverse of matrix $B$

This is a multiple choice questions, which I cant seem to understand. From what I do understand:

  1. Given two matrices $A$ and $B$, the product $AB$ is not equal to $BA$.
  2. Since they said $(A − B)(A + B) = A^2 − B^2$. It means that $AB$ is equal to $BA$ ie: they commute

Option 1 and Option 4 seems like the same term, but the left hand side is the terms are rearranged im not sure about them. I would choose option 3, since $A^{-1}A = I$ leaving $B=B$ is this correct?

Best Answer

$1.\ \ B = ABA^{-1}B \Leftrightarrow I = ABA^{-1} \Leftrightarrow A = AB \Leftrightarrow I = B$, so this is only the case when $B$ is the identity matrix.

$2.$ False, as we can only find out after doing the rest of the parts.

$3.\ \ B = A^{-1}BA \Leftrightarrow AB = BA$, which is something you have already shown is true.

$4.\ \ B = BABA^{-1} \Leftrightarrow BA = BAB \Leftrightarrow A = AB \Leftrightarrow I=B$, so again, this only holds when $B$ is the identity matrix.

$5.$ This is a solution, but not the only solution, since non-inverse matrices can commute. For example,

$$\begin{bmatrix}2&0\\1&-1\end{bmatrix}\begin{bmatrix}3&0\\1&0\end{bmatrix} = \begin{bmatrix}6&0\\2&0\end{bmatrix} = \begin{bmatrix}3&0\\1&0\end{bmatrix}\begin{bmatrix}2&0\\1&-1\end{bmatrix}, $$ but clearly the matrices aren't inverses.

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