# Can a matrix be just 1 number?

Can a matrix be just one number? Eg. Does 2 count as a matrix, if the question asks for a matrix to fit the question, but only the number 2 multiplying this particular matrix gives me the desired result?

Thanks

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Well, you could consider it a 1 by 1 matrix. – imranfat Feb 21 '14 at 16:37
Cheers.............. – Matt Feb 21 '14 at 16:51
What two matrices map the unit square, to coordinates (0,0) (2,0) (0,2) (2,2) ? – Matt Feb 21 '14 at 16:54
@Matt: Looks like you're looking for the matrix $2I$, or explicitly $\begin{pmatrix}2&0\\0&2\end{pmatrix}$. – Henning Makholm Feb 21 '14 at 17:07
I like your thinking. Often times, a "trivial case" like 1x1 matrices is just what you need to formulate a solution. – Jacob Wakem Feb 21 '14 at 17:08

Yes, a matrix with only one entry $c$, i.e., a $1 \times 1$ matrix, is equivalent to the scalar $c$. So any number in the underlying field can be thought of as a matrix with just one row, one column, i.e. one entry.

$$\begin{bmatrix} c\end{bmatrix} = c$$

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What two matrices map the unit square, to coordinates (0,0) (2,0) (0,2) (2,2) ? – Matt Feb 21 '14 at 16:50
One caveat: matrix multiplication of an $n\times m$ and a $p \times q$ matrix works iff $m = p$, but a scalar times a matrix always works. So $[\ c\ ] A \neq cA$, if $A$ has more than 1 row. – Emily Feb 21 '14 at 19:13

It seems that your problem is really a different one. It seems that you want to ask:

Can the map $\mathbb R^n\to\mathbb R^n$, $x\mapsto \lambda x$, where $\lambda\in\mathbb R$, also be expressed by a matrix $A$, i.e. is there a matrix $A$ such that $Ax=\lambda x$ for all $x\in\mathbb R^n$?

Yes, just let $A=\lambda I$, where $I$ is the unit matrix (all diagonal entries $1$, all other entries $0$).

If you already know what a linear map is, then one could also say:

Yes, since $x\mapsto\lambda x$ is linear and each linear map can be represented by a matrix.

And as a hint for your other problem: There are to linear maps that map the square spanned by the vectors $(0,1)$ and $(1,0)$ to itself, namely the identity and the reflection by the main diagonal.

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Yes, a number can be thought of as $1\times 1$ matrix. Note that for a general matrix $M$ (over, say, the real numbers) and a real number $a$, the notation $a.M$ is usually defined as the product $Ia. M$ where $I$ is the identity matrix, i.e. $a.M$ is obtained by multiplying all entries of $M$ with $a$.

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What two matrices map the unit square, to coordinates (0,0) (2,0) (0,2) (2,2) ? – Matt Feb 21 '14 at 16:53
The very last part seems wrong. – Carsten S Feb 21 '14 at 19:05
@CS, edited last part – fraction field Feb 21 '14 at 20:10