True or false? The nullity of an `n xx n` matrix might be any number from `0` to `n` .

Expert Answers
degeneratecircle eNotes educator| Certified Educator

This is true. It is easy to construct an `nxxn` matrix with any nullity from `0` to `n` inclusive. Simply take the `nxxn` ` ` matrix which contains the `kxxk` identity matrix in the upper left block and zeros everywhere else. This matrix will have nullity `n-k.` For the case of nullity `n` , simply take the zero matrix.

For example, a `4xx4` matrix with nullity `0` is `[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]].` A `4xx4` matrix with nullity `1` is `[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,0]],` and so on. The `4xx4` zero matrix has nullity `4`.