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# Linear algebra and matrix diagonalization

Please show all steps and workings clearly and explain in detail how you derive the answers.

1) This question concerns the linear transformation represented by a 3x3 matrix of real numbers.

With respect to the standard basis in both the domain and codomain.
(a) Determine the characteristic polynomial for A. [2]
(b) Show that lambda = 1 is an eigenvalue and find the remaining eigenvalues of A. [3]
(c) Determine all the corresponding eigenvectors and eigenspaces. [6]
(d) Write down a matrix P and a diagonal matrix D such that (P^-1)AP=D [3]
(e) Hence verify that (P^-1)AP=D by matrix multiplication. [6]

#### Solution Summary

Step by step, complete explanations are provided for answering each part of the given question on the properties of the example matrix linear transformation.

The discussion given throughout the solution should help with an understanding of the topics such as: solving the characteristic equation, finding eigenvectors and eigenvalues, and the diagonalisation of matrices. Care has been taken to explain all side topics required for the solution, such as the expansion of determinants and polynomial factorisation.

5 A4 pages in word document format.

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