Tagged: exam

Find the Rank of the Matrix $A+I$ if Eigenvalues of $A$ are $1, 2, 3, 4, 5$

Problem 35

Let $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix.

(UCB-University of California, Berkeley, Exam)

Read solution

LoadingAdd to solve later

The Subspace of Matrices that are Diagonalized by a Fixed Matrix

Problem 33

Suppose that $S$ is a fixed invertible $3$ by $3$ matrix. This question is about all the matrices $A$ that are diagonalized by $S$, so that $S^{-1}AS$ is diagonal. Show that these matrices $A$ form a subspace of $3$ by $3$ matrix space.

(MIT-Massachusetts Institute of Technology Exam)

Read solution

LoadingAdd to solve later