Question: 4. correctness, Let A be a n x n matrix. (1) Show that det(A) is the product of all eigenvalues of A. (2) Suppose that

4. correctness, Let A be a n x n matrix. (1) Show that det(A) is the product of all eigenvalues of A. (2) Suppose that A is diagonalizable. Show that tr( A) is the sum of all eigenvalues of A. (3) Set n = 2. Suppose that A is diagonalizable, and A has only nonnegative eigenval- ues. Show that tr(A) 2 det ( A ) n . n Remark. (2) is true for non-diagonalizable square matrices as well. Remark. (3) is true for all n > 0

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