How many of the following matrices have an eigenvalue 1? A = [1 0; 0 0], B =…

A matrix has eigenvalue 1 if det(A - I) = 0.

For A = [1 0; 0 0]:
A - I = [0 0; 0 -1], whose determinant is 0. So 1 is an eigenvalue.

For B = [0 1; 0 0]:
B - I = [-1 1; 0 -1], determinant = 1, so 1 is not an eigenvalue.

For C = [1 -1; 1 1]:
C - I = [0 -1; 1 0], determinant = 1, so 1 is not an eigenvalue.

For D = [-1 0; 1 -1]:
D - I = [-2 0; 1 -2], determinant = 4, so 1 is not an eigenvalue.

Therefore, exactly one matrix has eigenvalue 1.