# 1. Is the set of all 2 x 2 matrices of the form a 1/ 1 b , where a and b may be any scalars, a vector subspace of all 2 x 2 matrices?

Question
Matrices
1. Is the set of all 2 x 2 matrices of the form a 1/ 1 b , where a and b may be any scalars, a vector subspace of all 2 x 2 matrices?

2021-02-10
Step 1
The given set is $$\left\{ \left. \begin{bmatrix}a & 1 \\ 1 & b \end{bmatrix} \right| a,b \in K\right\}$$
It is known that any subspace should contain the identity element.
Step 2
Note that the identity matrix $$\begin{bmatrix}1 & 0 \\0& 1 \end{bmatrix}$$ does not belongs to $$\left\{ \left. \begin{bmatrix}a & 1 \\ 1 & b \end{bmatrix} \right| a,b \in K\right\}$$ Thus, the answer is NO. the set of all 2 x 2 matrices of the form $$\left\{ \left. \begin{bmatrix}a & 1 \\ 1 & b \end{bmatrix} \right| a,b \in K\right\}$$ is not a subspace of all 2 x 2 matrices.

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