Question

Q1. If you are given that ?-2? ? 1? is an eigenvector of A = ? 4 2 3? ? 0? ?-1 a -3?, ? 2 4 b ? (a) [5 marks] find the constants a and b in A; (b) [10 marks] construct a matrix P such that A = PDP?¹ with diagonal matrix D; (c) [5 marks] determine A is positive or negative definite or semidefinite, or indefinite, and explain. (d) [5 marks] deduce the eigenvalues and eigenvectors of C = A² - 2A + I, where I is the identity matrix. [Note: Do not attempt to compute the matrix C].

          Q1. If you are given that ?-2?
? 1? is an eigenvector of A = ? 4 2 3?
? 0? ?-1 a -3?,
 ? 2 4 b ?
(a) [5 marks] find the constants a and b in A;
(b) [10 marks] construct a matrix P such that A = PDP?¹ with diagonal matrix D;
(c) [5 marks] determine A is positive or negative definite or semidefinite, or indefinite, and explain.
(d) [5 marks] deduce the eigenvalues and eigenvectors of C = A² - 2A + I, where I is the identity matrix. [Note: Do not attempt to compute the matrix C].

Q1. If you are given that ?-2?
? 1? is an eigenvector of A = ? 4 2 3?
? 0? ?-1 a -3?,
? 2 4 b ?
(a) [5 marks] find the constants a and b in A;
(b) [10 marks] construct a matrix P such that A = PDP?¹ with diagonal matrix D;
(c) [5 marks] determine A is positive or negative definite or semidefinite, or indefinite, and explain.
(d) [5 marks] deduce the eigenvalues and eigenvectors of C = A² - 2A + I, where I is the identity matrix. [Note: Do not attempt to compute the matrix C].

Added by Megan B.

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