Can You Find a Matrix Where Column Space Equals Null Space?

[iNagib]

Hey fellas could use a hand on a linear question that has been bugging me.

Give an example of a 4 x 4 matrix such that col A = nul A
Appreciate any help!
Thanks,
ib

 

[adriank]

To start with, remember that dim(col A) + dim(null A) = 4, so we better have dim(col A) = dim(null A) = 2.

 

[adriank]

If col A = null A, then given any vector x in R⁴, Ax ∈ col A = null A, so A²x = 0. Since x is arbitrary, A² = 0. Conversely, if A² = 0, you can show that col A ⊆ null A, and you need dim(col A) = 2. This might help a bit in finding such a matrix.

(Matrices A such that A^k = 0 for some integer k are called nilpotent.)