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Q1. If you are given that $egin{pmatrix} -2 \ 1 \ 0 end{pmatrix}$ is an eigenvector of $A = egin{bmatrix} 4 & 2 & 3 \ -1 & a & -3 \ 2 & 4 & b end{bmatrix}$, (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 $C = A^2 - 2A + I$, where $I$ is the identity matrix. [Note: Do not attempt to compute the matrix $C$].

          Q1. If you are given that $egin{pmatrix} -2 \ 1 \ 0 end{pmatrix}$ is an eigenvector of $A = egin{bmatrix} 4 & 2 & 3 \ -1 & a & -3 \ 2 & 4 & b end{bmatrix}$,
(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 $C = A^2 - 2A + I$, where $I$ is the identity matrix. [Note: Do not attempt to compute the matrix $C$].

Q1. If you are given that eginpmatrix -2 1 0 endpmatrix is an eigenvector of A = eginbmatrix 4 2 3 -1 a -3 2 4 b endbmatrix,
(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 = A^2 - 2A + I, where I is the identity matrix. [Note: Do not attempt to compute the matrix C].

Added by Brent O.

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