some questions about the matrix

Question

Some questions about the matrix:
1. If $AB=BA$ that's imply A, B is square matrix?
2. is there equality $\det(2A)=8\det A$ for the matrix $A_{3x3}$?
3. If $\det A=0$ that's imply $Ax=b$ have no solution?

Answer 1

1. Yes. Suppose that $\newcommand{\reals}{{\bf R}}A\in\reals^{m\times n}$ and $B \in \reals^{p\times q}$. In order for $AB$ to be defined, it should be satisfied that $n=p$. Likewise, $q$ should equal to $m$ for $BA$ to be defined. Now since $AB\in\reals^{m\times q}$ and $BA\in\reals^{p\times n}$, $AB=BA$ implies that $m=p$ and $n=q$. Therefore, $AB=BA$ implies that $m=n=p=q$, thus, both $A$ and $B$ are square matrices (with the same dimensions).

2. Yes. When $A\in\reals^{n\times n}$ and $c\in\reals$, we have $\det(cA) = c^n \det(A)$. Refer to https://en.wikipedia.org/wiki/Determinant#Properties_of_the_determinant.

3. No, $\det(A)=0$ implies that $Ax=b$ has either no solution or infinite number of solutions. For example, if $A = b = 0$, there are infinite number of solutions. If $A=0$ and $b=1$, there is no solution.