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

Question

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?

Answer 1

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.

Answer 2

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.