# $2 \times 2$ matrices over {$0,1$} - linearly independent

I am a student in computer science - first year. I study linear algebra $$2$$ - course of linear algebra $$1$$ . In some institutions academic studies teach the courses together / teach in another way.

I tried to solve the question a few hours but I'm not sure how to solve it exactly.

the question is:

** Consider the $$2 × 2$$ matrices over {$$0, 1$$} whose rows are linearly independent.

a. Do they form a group under usual matrix addition?

b. Do they form a group if addition is carried out modulo $$2$$ ?

c. Do they form a group under matrix multiplication when summing the vector components is done modulo $$2$$ ?

which matrices are 'linearly independent'?

There are $$16$$ possible matrices - which ones are linearly independent?

They are referring to the non-singular matrix. For $$2 \times 2$$, the determinant of the matrix $$\begin{bmatrix} a & b \\ c & d\end{bmatrix}$$ is $$ad-bc$$.

Non-singular matrix has the property that $$ad-bc \ne 0$$.

Another properties is that the rows are not multiple of each other.

Hence suppose the first row is $$(1,0)$$, the second row can only be $$(0,1)$$ and $$(1,1)$$.

If first row is $$(0,1)$$, the second row can only be $$(1,0)$$ and $$(1,1)$$.

I will leave the case where the first row is $$(1,1)$$ to you.

• thank you - if its (1, 1) - so only two option are valid (0,1) and (1,0). – Haham May 5 at 8:02

The question mentioned that the rows of the matrix are linearly independent not the matrices. It's useful to take few examples on scratch. For example the rows of $$\begin{pmatrix}1&0\\0&1\\ \end{pmatrix}$$ are linearly independent. On the other hand, the rows of $$\begin{pmatrix}1&0\\0&0\\ \end{pmatrix}$$ and $$\begin{pmatrix}1&0\\1&0\\ \end{pmatrix}$$ are linearly dependent.