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I want to find all of the units of the ring $M_{2} (\mathbb{Z}_2)$ (the set of $2\times 2$ matrices with entires from $\mathbb{Z}_2$)

I have tried to solve it by taking two matrices and labeling the entries $a,b,c,d$ and $w,x,y,z$ respectively. Then if you multiply them together you get:

\begin{array}{cc} aw+by & ax+bz \\ cw+dy & cx+dz \\ \end{array}

this then gives me the following requirements:

$aw+by\equiv cx+dz \equiv 1$ and $ zx+bz \equiv cw+dy \equiv 0$ (mod 2)

From this I can show any matrix of the form

\begin{array}{cc} 0 & 0\\ \alpha & \beta \\ \end{array}

or

\begin{array}{cc} \alpha & \beta \\ 0 & 0 \\ \end{array}

cannot be a unit $\forall \alpha, \beta \in \mathbb{Z}_2$

Do I just brute force the rest of the possible matrices? Or am I missing something?

I would really just like a hint not the solution.

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  • $\begingroup$ Do you know how to solve this in a more typical setting? For example, do you know which matrices in $M_2(\Bbb R)$ are units? $\endgroup$ Commented Oct 11, 2022 at 18:02
  • $\begingroup$ Also, do you know the formula for the inverse of a $2\times 2$ matrix? $\endgroup$ Commented Oct 11, 2022 at 18:04
  • $\begingroup$ Alternatively: a matrix is invertible if and only if its rows are linearly independent. There aren't that many nonzero vectors in $(\mathbb{Z}_2)^2$, and since two vectors are linearly independent if and only if neither a scalar multiple of the other, you can easily list all the answers. $\endgroup$ Commented Oct 11, 2022 at 18:07

2 Answers 2

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A matrix in $M_2(\mathbb Z_2)$ is a unit if and only if its determinant is equal to $1$. This should give a very simple way to find all units (there is not much to check).

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Here is a hint for a strategy to solve a wide class of problems, that has been very fruitful to me:

(1) Ignoring whether or not the matrix is invertible, what would the general $2\times 2$ matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ look like, if $a,b,c,d$ are REAL numbers?

(2) Notice that in your answer to (1), there is an operation that is not always allowed, but if it was allowed, then their would be no obstruction to the existence of the inverse. What is this operation, and can you deduce what a necessary and sufficient condition is for a matrix with REAL entries to be invertible?

(3) Same questions as (1) and (2), but this time over $\mathbb{Z}_2$, that is, find a general form of what would be the inverse if it were to exist, then find the obstruction (HINT: division by 0 in $\mathbb{Z}_2$), then deduce that if this obstruction doesn't hold, the matrix is invertible.

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