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9. If C1, C2 are distinct eigenvalues of a 2x2 matrix A corresponding respectively to eigenvectors U1, Uz then we say that A = Cjuju1 + Czuzu2 is the Spectral Decomposition of A. Find the spectral decomposition of A = 4 2] 10. Suppose that A be a square, invertible matrix such that A3 = A and 2. is a positive eigenvalue of . Find the value of 2. me anything 0 0 92. Suppose 42% of Ph.D.s In statistics/biostatistics are earned by women and a random sample 36 statistics/blostatistics Ph.D.s is obtained. (You may want to use technology to compute some of the probabilities.) a. What's the probability none of the 36 statistics/blostatistics Ph.D.s are women? b. What's the probability at least 1 of the 36 statistics/biostatistics Ph.D.s is a woman? C. How many women do you expect in the sample of the 36 statistics/biostatistics Ph.D.s? d. What's the probability fewer than half of the 36 statistics/biostatistics Ph.D.s are women?2 1. Given the matrix A = 8 i) Calculate the eigenvalues X, and A2 together with the associated eigenvectors x, and x2 ii) determine the spectral decomposition of A iii) find A-1 2. Given the following 3 x 3 symmetric matrix A, A = - 2 1 i) find the eigenvalues and the associated normalized eigenvectors of A. ii) find tr( A) and [A | and show that tr(A) = )., A, and JA| = II, A. ili) write the matrix A in its spectral decomposition form.According to a study carried out by the firm Optima Groupe Conseil for the newspaper Les Affaires, the average income per household in the Trois-Rivieres-Quest geographic area is $ 55,577 (this value was obtained from the Statistics Canada database - 2001 census). Let us assume that the standard deviation of the income per household is $ 4,000. A random sample of 64 households in this geographic area is collected. a) What are the parameters of the sampling distribution of the variable "average income per household" for a sample of size n = 64? (1 point) b) What assumptions and conditions are required to answer c) and d)? (1 point) c) What is the probability that the average income in the sample per household in the Trois- Rivieres-Quest geographic area will be less than $ 54,777? (2 points) d) What is the probability that the average income in the sample per household in the Trois- Rivieres-Quest geographic area is between $ 54,327 and $ 56,827? (2 points)