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ECE-GY 5253 Midterm
Fall 2022
Due: Monday, October 31, 7:30 pm (New York Time)
1 Problem 1
Are the following statements true or false? If true, prove the statement. If false, give a counterexample.
1. Let A ∈ Rn×n. If A2 = A, then eigenvalues of A are either 0 or 1.
2. Let A ∈ Rm×n. If ATA = 0, then A = 0. (Note: 0 ∈ Rm×n denotes zero matrix with only zeros
as its elements)
3. Let A,B ∈ Rn×n be two real symmetric matrices. If xTAx = xTBx for all x ∈ Rn, then A = B.
2 Problem 2
If A ∈ Rn×n has n distinct eigenvalues and commutes with a given matrix B ∈ Rn×n, show that
1. B is diagonalizable.
2. There exist coefficients a0, a1, ..., an−1 such that
B = an−1An−1 + an−2An−2 + ...+ a1A+ a0I
where I is the identity matrix.
3 Problem 3
For the following matrix A =
−2 2 1−7 4 2
5 0 0
, is the matrix A diagonalizable? If yes, explain. If not,
what is its Jordan form?