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i) Show that if det A = 0, then A has an eigenvalue of 0. ii) Show that if A has an eigenvalue of 0, then det A = 0. iii) Let T :R² → R? be the linear transformation corresponding to the 2 x 2 matrix A. Suppose A has an eigenvalue of 0, corresponding to the eigenvector (). Geometrically describe the effect of this transformation on areas in 2-dimensional Euclidean space.

i) Show that if det A = 0, then A has an eigenvalue of 0. ii) Show that if A has an eigenvalue of 0, then det A = 0. iii) Let T :R² → R? be the linear transformation corresponding to the 2 x 2 matrix A. Suppose A has an eigenvalue of 0, corresponding to the eigenvector (). Geometrically describe the effect of this transformation on areas in 2-dimensional Euclidean space.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let A be a 2 × 2 matrix. i) Show that if det A = 0, then A has an eigenvalue of 0. ii) Show that if A has an eigenvalue of 0, then det A = 0. iii) Let T: R? → R? be the linear transformation corresponding to the 2 × 2 matrix A. Suppose A has an eigenvalue of 0, corresponding to the eigenvector (). Geometrically describe the effect of this transformation on areas in 2-dimensional Euclidean space.
Transcribed Image Text:Let A be a 2 × 2 matrix. i) Show that if det A = 0, then A has an eigenvalue of 0. ii) Show that if A has an eigenvalue of 0, then det A = 0. iii) Let T: R? → R? be the linear transformation corresponding to the 2 × 2 matrix A. Suppose A has an eigenvalue of 0, corresponding to the eigenvector (). Geometrically describe the effect of this transformation on areas in 2-dimensional Euclidean space.
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