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Unformatted text preview: MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 9 1. A matrix S ∈ R n × n is called skew symmetric if S > =- S . (a) For any matrix A ∈ R n × n for which I + A is nonsingular, show that ( I- A )( I + A )- 1 = ( I + A )- 1 ( I- A ) . (1.1) We will write I- A I + A for the matrix in (1.1). [Note: In general, AB- 1 6 = B- 1 A and so A B is ambiguous since it could mean either AB- 1 or B- 1 A .] (b) Let Q ∈ R n × n be an orthorgonal matrix such that I + Q is nonsingular. Show that I- Q I + Q is a skew symmetric matrix. (c) Let S ∈ R n × n be a skew symmetric matrix. Show that I- S I + S is an orthogonal matrix. (d) Why is it unnecessary to require that I + S be nonsingular in (c)? [Hint: Problem 3 below.] 2. Let A,B ∈ R n × n . Let λ a ∈ R be an eigenvalue of A and λ b ∈ R be an eigenvalue of B . (a) Is it always true that λ a λ b is an eigenvalue of AB ? Is it always true that λ a + λ b is an eigenvalue of A + B ?...
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