The geometric mutiplicity of a matrix

[chuy52506]

It is a 5x5 matrix with 1s in all of its entries.
Find the geometric multiplicity of $$\lambda$$=0 as an eigenvalue of the matrix.

 

[HallsofIvy]

What have you done yourself?

Solve the equation
$$\begin{bmatrix}1 & 1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 & 1 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5\end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \\ 0 \\ 0\end{bmatrix}$$.

How many distinct equations do you get relating those variables? Can you solve for some in terms of the others?

 

[Simon_Tyler]

Here's the text of a http://www.sagenb.org/" that answers your question.
Since you've shown no work, I'll leave the interpretation up to you.

sage: A = Matrix(QQ,[[1,1,1,1,1],[1,1,1,1,1],[1,1,1,1,1],[1,1,1,1,1],[1,1,1,1,1]]); A
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]

sage: A.eigenvalues()
[5, 0, 0, 0, 0]

sage: A.eigenspaces()
[
(5, Vector space of degree 5 and dimension 1 over Rational Field
User basis matrix:
[1 1 1 1 1]),
(0, Vector space of degree 5 and dimension 4 over Rational Field
User basis matrix:
[ 1  0  0  0 -1]
[ 0  1  0  0 -1]
[ 0  0  1  0 -1]
[ 0  0  0  1 -1])
]