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Question 2.7.17: Find 2 by 2 symmetric matrices A = A^T with these properties...

Find 2 by 2 symmetric matrices A = {A}^{T} with these properties:
(a) A is not invertible.
(b) A is invertible but cannot be factored into LU (row exchanges needed).
(c) A can be factored into LD{L}^{T} but not into L{L}^{T}(because of negative D).

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(a) A=\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}={A}^{T} is not invertible

(b) A=\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix} needs row exchange

(c) A=\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} has D=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.

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