Question

4.Is it sometimes, always, or never true that, if A and B both have dimensions n x n , AB=BA ? Explain.

Answer 1

Answer: Is true sometimes.

Step-by-step explanation:

I guess that here we have two matrices, A and B, that are nxn.

We can see that if those matrices can conmutate, then we can try it with some simple matrices.

$A = \left[\begin{array}{ccc}1&0\\0&-1\end{array}\right] . B = \left[\begin{array}{ccc}2&0\\1&1\end{array}\right]$

Here, we would have that:

$AB = \left[\begin{array}{ccc}2&0\\-1&-1\end{array}\right]$

$BA = \left[\begin{array}{ccc}2&0\\1&-1\end{array}\right]$

You can see that AB and BA are different, then the statement is not always true.

But it is true sometimes, if A or B are the identiti, then I*A = A*I, in this case would be true.

It is also true if A and B are diagonal matrices, let's prove it:

$A = \left[\begin{array}{ccc}a&0\\0&b\end{array}\right] , B = \left[\begin{array}{ccc}c&0\\0&d\end{array}\right]$

$AB = \left[\begin{array}{ccc}ac&0\\0&bd\end{array}\right] = BA$