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Exercise 7.6.9

Avg difficulty 3.0

Consider the matrix

A=[ete−tcos⁡te−tsin⁡tet−e−tcos⁡t−e−tsin⁡t−e−tsin⁡t+e−tcos⁡tet2e−tsin⁡t−2e−tcos⁡t].A=\left[\begin{array}{ccc}e^{t}&e^{-t}\cos t&e^{-t}\sin t\\ e^{t}&-e^{-t}\cos t-e^{-t}\sin t&-e^{-t}\sin t+e^{-t}\cos t\\ e^{t}&2e^{-t}\sin t&-2e^{-t}\cos t\end{array}\right].

Does there exist a value of tt for which this matrix fails to be invertible? Explain.

Selinger, Matrix Theory and Linear Algebra, Determinants, Application: A formula for the inverse of a matrix, Exercise 7.6.9

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.