Are the image and kernel of a 3x3 matrix ever equal If so give an example?

Answer 1

No.

A 3x3 matrix A is a representation of a linear map $\backslash alpha\; :\; \backslash mathbb\{R\}^3\; \backslash longrightarrow\; \backslash mathbb\{R\}^3$.

For any linear map $T\; :\; U\; \backslash longrightarrow\; V$,

we have the rank-nullity theorum:

rank(T)+nullity(T) = dim(U)

where the rank and nullity are the dimensions of the image and kernal of T respectively.

Im(T) = ker(T) $\backslash Rightarrow$ rank(T) = nullity(T) = m, say

for some non-negative integer m. Then the rank-nullity theorum implies that dim(U)=2m.

The image and kernal of a matrix A are the same as those for the corresponding basis-free linear map $\backslash alpha$.

For a 3x3 matrix, dim(U) = 3, so there are no such matrices (since 3 is odd).