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1 If and are two matrices, prove the following properties of the trace of a matrix..., for a any constant .

2 If and are two matrices, prove the following properties of the determinant of a matrix.det = det .det = det det = det .

3 LetFind .Find .Find .Find .

4 LetFind .Find .Compare and .

5 LetFind .Find .

6 Show that the real symmetric matrixis positive definite for any non‐zero column vector.

7 Prove that if and are positive definite matrices then so is .

8 For what values of is the following matrix positive semidefinite?

9 Decide whether the following matrices are positive definite, negative definite, or neither. Please explain your reasoning.

10 For random variables and , show thatThe variance is the variance of the random variable , while the same holds for the random variable .