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
asked 2021-02-09
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?

Answers (1)

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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