Question

How many of the following matrices have an eigenvalue 1? [[1,0],[0,0]], [[0,1],[0,0]], [[1,−1],[1,1]], and [[−1,0],[1,−1]]

Accepted Answer

The correct answer is (A) one. [[1,0],[0,0]] is diagonal with eigenvalues 1 and 0 — HAS eigenvalue 1. [[0,1],[0,0]] has characteristic equation λ²=0, eigenvalues 0,0 — no. [[1,−1],[1,1]] has trace 2, det 2, characteristic equation λ²−2λ+2=0, giving complex eigenvalues 1±i — not the real value 1. [[−1,0],[1,−1]] is lower-triangular with both diagonal entries −1, so both eigenvalues are −1 — no. Only the first matrix has eigenvalue 1, so the count is one.

Explanation