2.II.10C

Linear Algebra | Part IB, 2005

(i) Let A=(aij)A=\left(a_{i j}\right) be an n×nn \times n matrix with entries in C. Define the determinant of AA, the cofactor of each aija_{i j}, and the adjugate matrix adj(A)\operatorname{adj}(A). Assuming the expansion of the determinant of a matrix in terms of its cofactors, prove that

adj(A)A=det(A)In\operatorname{adj}(A) A=\operatorname{det}(A) I_{n}

where InI_{n} is the n×nn \times n identity matrix.

(ii) Let

A=(0100001000011000)A=\left(\begin{array}{llll} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right)

Show the eigenvalues of AA are ±1,±i\pm 1, \pm i, where i2=1i^{2}=-1, and determine the diagonal matrix to which AA is similar. For each eigenvalue, determine a non-zero eigenvector.

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