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

Let $K$ be a field, and let

$T=\{M\in {\mathrm{Mat}}_{3\times 3}(K)\mid detM=0\}\subset {\mathrm{Mat}}_{3\times 3}(K)=K^9$

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

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

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

Create a solution