Choice of variable to make matrix rank = 1

[gtse]

Homework Statement

What choice of d would make matrix
a b
c d
have a rank of 1?

Homework Equations

rank(A) + nullity(A) = n

The Attempt at a Solution

In order for rank = 1, then nullity must = 1 because n = 2.
This isn't a nonsingular matrix, so det(A) = ad-bc =/= 0.
d = bc/a, where 'a' cannot be 0.
Let's say a = 1, then the matrix will be
1 ad/c
c d
c cannot be 0 either, so matrix is now
1 d
1 d
and I guess a possible candidate would just then be
1 1
1 1?

I think I'm supposed to give the answer in more arbitrary terms, so I am lost here.

 

[mighty2000]

Hello, thank you for your post.

You are correct in your approach that in order for the rank of the matrix to be 1, the nullity must also be 1. This means that there is only one linearly independent row or column in the matrix.

To find a more general solution, we can let d = k, where k is a constant. This means that the matrix will be
a b
c k

If we let a = 1 and c = 0, then the matrix would be
1 b
0 k
This matrix has a rank of 1 because the first row is linearly independent, and the second row is a multiple of the first row.

In general, any matrix of the form
a b
0 k
will have a rank of 1 as long as a and k are not both 0.

I hope this helps! Let me know if you have any other questions.