TF. In parts (a)(l) determine whether the statement is true or false, and justify your

Chapter 7, Problem T2

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TF. In parts (a)(l) determine whether the statement is true or false, and justify your answer. (a) If all eigenvalues of a symmetric matrix A are positive, then A is positive definite. (b) x2 1 x2 2 + x2 3 + 4x1x2x3 is a quadratic form. (c) (x1 3x2)2 is a quadratic form. (d) A positive definite matrix is invertible. (e) A symmetric matrix is either positive definite, negative definite, or indefinite. (f ) If A is positive definite, then A is negative definite. (g) x x is a quadratic form for all x in Rn. (h) If A is symmetric and invertible, and if xT Ax is a positive definite quadratic form, then xT A1x is also a positive definite quadratic form. (i) If A is symmetric and has only positive eigenvalues, then xT Ax is a positive definite quadratic form. ( j) If A is a 2 2 symmetric matrix with positive entries and det(A) > 0, then A is positive definite. (k) If A is symmetric, and if the quadratic form xT Ax has no cross product terms, then A must be a diagonal matrix. (l) If xT Ax is a positive definite quadratic form in two variables and c = 0, then the graph of the equation xT Ax = c is an ellipse. Use the eigenvalues of the following matrix to determine whether it is positive definite, negative definite, or idefinite, and then confirm your conclusion using Theorem 7.3.4. A = 5 3030 3 2020 0 0 111 321 8 2 0012 7

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