Chapter 6 - Vector Spaces - 6.2 Linear Independence, Basis, and Demension - Exercises for 6.2 - Page 457: 21

$\mathcal{B}$ is not a basis of $M_{22}$.

The vector space $M_{22}$ has dimension $4$. So, any set of vectors of cardinality greater than $4$ must be linearly dependent. Since $\mathcal{B}$ has $5$ elements, $\mathcal{B}$ cannot a basis of $M_{22}$.