Question

True or false: (a) If A2 is defined then A is necessarily square. (b) If AB and BA are defined then A and Bare square. (c) If AB and BA are defined then AB and BA are square. (d) If AB

Answer 1

(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

$A_{3\times2} = \begin{bmatrix}1&0\\0&1\\0&0\end{bmatrix}$

$B_{2\times3} = \begin{bmatrix}-1&0&1\\0&0&1\end{bmatrix}$

Then both AB and BA are defined, with

$AB_{3\times3} = \begin{bmatrix}-1&0&1\\0&0&1\\0&0&0\end{bmatrix}$

$BA_{2\times2} = \begin{bmatrix}-1&0\\0&0\end{bmatrix}$

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.