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?

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?
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Talisha
Step 1
The given set is $\left\{\left[\begin{array}{cc}a& 1\\ 1& b\end{array}\right]|a,b\in K\right\}$
It is known that any subspace should contain the identity element.
Step 2
Note that the identity matrix $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ does not belongs to $\left\{\left[\begin{array}{cc}a& 1\\ 1& b\end{array}\right]|a,b\in K\right\}$ Thus, the answer is NO. the set of all 2 x 2 matrices of the form $\left\{\left[\begin{array}{cc}a& 1\\ 1& b\end{array}\right]|a,b\in K\right\}$ is not a subspace of all 2 x 2 matrices.
Jeffrey Jordon