(a) True. Suppose A is a not a square matrix, with m rows and n columns. Then A² is not defined, because you can't multiply an m×n matrix by another m×n matrix.
(b) False. As an example, consider the matrices
[tex]A_{3\times2} = \begin{bmatrix}1&0\\0&1\\0&0\end{bmatrix}[/tex]
[tex]B_{2\times3} = \begin{bmatrix}-1&0&1\\0&0&1\end{bmatrix}[/tex]
Then both AB and BA are defined, with
[tex]AB_{3\times3} = \begin{bmatrix}-1&0&1\\0&0&1\\0&0&0\end{bmatrix}[/tex]
[tex]BA_{2\times2} = \begin{bmatrix}-1&0\\0&0\end{bmatrix}[/tex]
In general, you can multiply any m×n by any n×m matrix.
(c) True. Multiplying a m×n matrix by a n×m matrix always yields a m×m matrix, and multiplying a n×m matrix by a m×n matrix always yields a n×n matrix.
