What does this notation $J = 11^T ∈ M_n(R)$ mean?

Question

I am solving some problems in Matrices and I came across this problem, I don't want the solution. I am having a little difficulty in understanding the notation given in the problem. Here is the problem,

Let $J = 11^T ∈ M_n(R)$. Then each entry of $J$ equals $1$. Which among these is the incorrect option?

And he gave some options. Can somebody explain what does the question mean? I don't understand what is this $11^T$ and what is $\mathbb{M}_n(\mathbb{R})$?

If writing down the options can help, I can add those options in the question.

Answer 1

From context, it can be inferred that $1$ is the $n\times 1$ matrix with all entries equal to $1$, i.e. $$ 1 = \begin{pmatrix}1\\ 1\\ \vdots\\ 1\end{pmatrix}. $$ Given a matrix $A$, the matrix $A^T$ denotes the transpose of that matrix. That is, if $a_{ij}$ denotes the entry in the $i$-th row and $j$-th column of $A$, then the value in the $i$-th column and $j$-th row of $A^T$ is $a_{ij}$, i.e. $$ (a^T)_{ji} = a_{ij}. $$ Hence $$ 1^T = \begin{pmatrix} 1 & 1 & \dotsb & 1\end{pmatrix}.$$ Again assuming that $1$ is of length $n$, it therefore follows that $$ J = 11^T = \begin{pmatrix}1\\ 1\\ \vdots\\ 1\end{pmatrix}\begin{pmatrix} 1 & 1 & \dotsb & 1\end{pmatrix} = \begin{pmatrix} 1 & 1 & \dotsb & 1 \\ 1 & 1 & \dotsb & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \dotsb & 1 \\ \end{pmatrix}, $$ i.e. the $n\times n$ matrix consisting entirely of $1$s.

Finally, (again inferring from context, though this notation is reasonably standard) $\mathbb{M}_{n}(\mathbb{R})$ is the set of all $n\times n$ matrices with real entries. For example, $\mathbb{M}_2(\mathbb{R})$ consists of all matrices of the form $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ where $a$, $b$, $c$, and $d$ are all real numbers.

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On a slightly different topic, I feel like the notation in this problem was written to confound, rather than illuminate (though, as I said, we can infer the correct meaning from context). I might have preferred to write something like

Let $\mathbb{1}_n$ denote the $n\times 1$ matrix $$ \mathbb{1}_n = \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1\end{pmatrix}. $$ Then $J = \mathbb{1}_n \cdot \mathbb{1}_n^T \in \mathbb{M}_n(\mathbb{R})$ is the $n\times n$ matrix in which every entry is $1$.

I think that it vastly improves readability to use a different font for this $\mathbb{1}_n$ matrix, and to explicitly indicate its size in the notation.