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.

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.