course |
course_year |
question_number |
tags |
title |
year |
Linear Algebra |
IB |
3 |
|
2.II.10C |
2005 |
(i) Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix with entries in C. Define the determinant of $A$, the cofactor of each $a_{i j}$, and the adjugate matrix $\operatorname{adj}(A)$. Assuming the expansion of the determinant of a matrix in terms of its cofactors, prove that
$$\operatorname{adj}(A) A=\operatorname{det}(A) I_{n}$$
where $I_{n}$ is the $n \times n$ identity matrix.
(ii) Let
$$A=\left(\begin{array}{llll}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0
\end{array}\right)$$
Show the eigenvalues of $A$ are $\pm 1, \pm i$, where $i^{2}=-1$, and determine the diagonal matrix to which $A$ is similar. For each eigenvalue, determine a non-zero eigenvector.