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MATH 70095 Multivariate Statistical Methods

Q1) For the following matrices,
• Calculate all eigenvalues and eigenvectors;
• Find the geometric and algebraic multiplicities of each eigenvalue, and use these to establish whether the matrix can be diagonalised;
• If it exists, state the diagonalisation of the matrix;
• Provide a geometric interpretation of the linear transformation(s) represented by the matrix.
For part (b), you should consider all possible matrices for a, b ∈ R, a 6= 0, b 6= 0. [15]

Q2) Suppose we roll two (possibly weighted) six-sided dice independently, and we are interested in whether one or both dice land with an even number facing upwards (i.e. whether they ‘show’ an even number). Let E1 denote the event that the first dice

shows an even number, and let E2 denote the event that the second dice shows an even number. Furthermore, let

qEE := P(both dice show an even number),
qEO := P(exactly one dice shows an even number),
qOO := P(both dice show an odd number).

If P(E1) = a and P(E2) = b, find a, b such that qOO = qEO = qEE/2, or show that this is not possible.