Let $\mathbf{A}, \mathbf{B}, \mathbf{C}$ and $\mathbf{D}$ be the following matrices: \begin{align*} &\mathbf{A} \text{ sends every vector to the zero vector}, \\ &\mathbf{B} \text{ reflects vectors across the $xz$-plane}, \\ &\mathbf{C} \text{ projects vectors onto the vector $\mathbf{i}+ 3 \mathbf{k}$}, \\ &\mathbf{D} \text{ sends every vector to twice itself}. \end{align*}For each matrix above, figure out whether it's invertible or not.
If I have an actual matrix, I know how to find if it's invertible or not using the determinant. Without and actual matrix, I'm not too sure how to figure out if it's invertible or not. Could someone help me please?
Don't see them as matrices. See them as linear maps instead. Do these linear maps have inverses. For instance, D does: its inverse is the map that maps each vector into half of it. And A doesn't, since it is not injective.