THEUNIVERSITY OFSYDNEYPUREMATHEMATICSLinear Mathematics2012Tutorial 51.Consider the following two matricesA=parenleftBig2 0-20 3 00 0 3parenrightBigandB=parenleftBig1 0 00 1 10 1 1parenrightBig.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 ofR3consisting of eigenvectors of the matrix.c)Write down the square matrixPwhose columns are the basis vectors you found in part b).(Such a matrix is invertible. Why?) Write down the diagonal matrixDwhose diagonalentries are the eigenvalues of the matrix, in the same order as the corresponding columnsofP.d)Confirm that the matrix is diagonalisable by performing appropriate matrix multiplica-tions.2.LetM=parenleftBig4-2 12 0 12-2 3parenrightBig. Show thatMis diagonalisable, and findMn, forn≥1.3.Determine whether or not the following matrices are diagonalisable.
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