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# tut05 - T HE U NIVERSITY OF S YDNEY P URE M ATHEMATICS...

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T HE U NIVERSITY OF S YDNEY P URE M ATHEMATICS Linear Mathematics 2012 Tutorial 5 1. Consider the following two matrices A = parenleftBig 2 0 - 2 0 3 0 0 0 3 parenrightBig and B = parenleftBig 1 0 0 0 1 1 0 1 1 parenrightBig . For each of these matrices: a ) Find all the eigenvalues for the matrix and, for each eigenvalue, find a basis for the corre- sponding eigenspace. b ) Find a basis of R 3 consisting of eigenvectors of the matrix. c ) Write down the square matrix P whose columns are the basis vectors you found in part b). (Such a matrix is invertible. Why?) Write down the diagonal matrix D whose diagonal entries are the eigenvalues of the matrix, in the same order as the corresponding columns of P . d ) Confirm that the matrix is diagonalisable by performing appropriate matrix multiplica- tions. 2. Let M = parenleftBig 4 - 2 1 2 0 1 2 - 2 3 parenrightBig . Show that M is diagonalisable, and find M n , for n 1 . 3. Determine whether or not the following matrices are diagonalisable.
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