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Unformatted text preview: Math 304 Solutions 7 1 Section 6.1 4. Let A be a nonsingular matrix and let be an eigenvalue of A . Show that 1 / is an eigenvalue of A- 1 . Answer: Let v be the eigenvector for A corresponding to the eigen- value . Geometrically, the result of multiplying a multiple of v by A is to multiply its length by . Thus, multiplying by A- 1 will multiply the length of the vector by 1 / . Here is a more algebraic proof: Consider A- 1 ( A v ): A- 1 ( A v ) = v Since A v = v , A- 1 ( A v ) = A- 1 ( v ) Thus, we have that A- 1 ( v ) = v Thus, v is an eigenvector for A with eigenvalue 1 / . 10. Show that the matrix A = cos - sin sin cos will have complex eigenvalues if is not a multiple of . Give a geo- metric interpretation of this result. Answer: det( I- A ) = ( - cos ) 2 + sin 2 1 If det( I- A ) = 0, then ( - cos ) 2 =- sin 2 If is real, the left side of this equation is always greater than or equal to 0, and the right side of this equation is always less than or equal to...
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This note was uploaded on 06/11/2010 for the course MA 405 taught by Professor Staff during the Spring '08 term at N.C. State.
- Spring '08