Find a nonzero matrix?

Question

Find a nonzero matrix A for which A^2=0.

 

Please help me with steps. Thanks.

Answer 1

Let A be a 2x2 matrix.

|a b||a b|

|c d||c d|

=

|a^2+bc ab+bd|

|ac+cd bc+d^2|

 

If the product is a zero matrix, you must have

a = d = 0, and either b = 0 or c = 0.

Answer for 2x2 matrix:

|0 b|

|0 0|

or

|0 0|

|c 0|

where b and c can be any non-zero number.

Answer 2

The 2x2 matrix

[0  1]

[0  0]

is the simplest example of such a matrix. Look up nilpotent matrices for more information.