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Let there be a set $$ \mathscr M⊆\mathbf M_3^\Bbb C,\mathscr M≠∅ $$ with the following properties $$A,B∈ \mathscr M\Longrightarrow A+B∈ \mathscr M $$ $$ A∈\mathscr M,C∈M_3^\Bbb C\Longrightarrow CA∈\mathscr M $$ $$ X∈M_{3,1}^\Bbb C,AX=O,\forall A∈\mathscr M \Longrightarrow X=0.$$ Prove that $$ \mathscr M=\mathbf M_3^\Bbb C.$$

user26857
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Razvan Paraschiv
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  • What does $M_{3,1}^\Bbb C$ denote? And is $M_3^\Bbb C$ the set of $3\times 3$ matrices? – Arnaud D. Nov 06 '16 at 20:37
  • @ArnaudD. $\mathbf M_{3,1}^\Bbb C$ represents the set of matrices with 3 lines and 1 column. For the second question, yes, it represents the set of 3x3 matrices. – Razvan Paraschiv Nov 06 '16 at 20:49

2 Answers2

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Let $E_{i,j}$ be a matrix consisting of all zeroes except for a $1$ on the $i$-th row and the $j$-th column. This matrix has the property that for another matrix $M$ the product $E_{i,j}M$ has all zeroes except for the $i$-th row of $M$ that shows up at the $j$ -th row. From this we conclude that the matrices of $\mathscr M $ are of the form $\begin{pmatrix}u_1 & u_2 & u_3\\v_1 & v_2 & v_3\\ w_1 & w_2 & w_3 \end{pmatrix}$ where $(u_1,u_2,u_3), (v_1,v_2,v_3), (w_1,w_2,w_3)$ are rows in a subspace $W $ of $\mathbf M_{1,3}^\Bbb C$, the set of $1 \times 3$ matrices. If $W$ has dimension $3$ we're done. If not then there exists a non-zero column $X = \begin{pmatrix}z_1 \\ z_2 \\ z_3 \end{pmatrix}$ such that $AX = 0$ $ \forall A \in \mathscr M $, but this is in contradiction with the last property.

Marc Bogaerts
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  • Can you please adapt your solution without using vectors. Im a first year university student and i havent learned them yet.i have understood everything till the line you used "W". – Razvan Paraschiv Nov 07 '16 at 22:19
  • In addition , why do you take only the fundamental matrices out of the whole set of matrices with complex elements? – Razvan Paraschiv Nov 07 '16 at 23:12
  • I edited my answer accordingly. The funcamental matrices are used to filter out elements of $\mathscr M$ that consist of all zeroes except for one row. You can even switch rows from place, showing that all the rows belong to the same subspace of rows. Play a bit around with concrete examples and you will see what happens. – Marc Bogaerts Nov 07 '16 at 23:21
  • I apologise for bothering you with some many(probably juvenile)questions,but i was wondering why am i given the first property and ,secondly, i wanted to ask if u1,u2,u3 represent the coefficients of i,j,k in the group pf quaternions without real part. – Razvan Paraschiv Nov 07 '16 at 23:41
  • No problem. Nothing to do with quaternions, they're just arbitrary complex numbers. I could have written $(m_{1,1}, m_{1,2},m_{1,3})$ and $(m_{2,1}, m_{2,2},m_{2,3})$ etc. but I was a bit lazy. – Marc Bogaerts Nov 07 '16 at 23:50
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Let $M_3(\mathbb{C})$ be the set of $3 \times 3$ matrices over $\mathbb{C}$. Note that your first two conditions on $M$ together with $M \neq \emptyset$ are telling you that $M$ is a left ideal in $M_3(\mathbb{C})$. Such ideals can be described, see for example What are the left and right ideals of matrix ring? How about the two sided ideals?. Now combine this with your last condition on $M$ to finish the proof.

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