3x3-matrices/Determinant 0/Linear subspace/Dimension 6/Exercise

Let ${}K$ be a field, and let

${}T={\left\{M\in \operatorname {Mat} _{3\times 3}(K)\mid \det M=0\right\}}\subset \operatorname {Mat} _{3\times 3}(K)=K^{9}\,$

be the set of all ${}3\times 3$-matrices with determinant ${}0$.

a) Show that ${}T$ is not a linear subspace of ${}\operatorname {Mat} _{3\times 3}(K)$.

b) Show that ${}T$ contains a linear subspace of ${}\operatorname {Mat} _{3\times 3}(K)$ of dimension ${}6$.

Create a solution