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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?

vmahth1
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  • Hello! Please add some context to this question with an [edit]. Where do these problems come from? What are your thoughts? What have your tried so far, and where did that fail? – Brian61354270 Feb 16 '20 at 00:41
  • I have no idea, that's why I'm asking – vmahth1 Feb 16 '20 at 00:43
  • You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Brian61354270 Feb 16 '20 at 00:44

1 Answers1

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  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.

Sunghee Yun
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