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Question 2.7.21: After elimination clears out column 1 below the first pivot,...

After elimination clears out column 1 below the first pivot, find the symmetric 2 by 2 matrix that appears in the lower right comer:
Start from A =\left[ \begin{matrix} 2 & 4 & 8 \\ 4 & 3 & 9 \\ 8 & 9 & 0 \end{matrix} \right] and A =\left[ \begin{matrix} 1 & b & c \\ b & d & e \\ c & e & f \end{matrix} \right].

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Elimination on a symmetric 3 by 3 matrix leaves a symmetric lower right 2 by 2 matrix. 

The examples \left[ \begin{matrix} 2 & 4 & 8 \\ 4 & 3 & 9 \\ 8 & 9 & 0 \end{matrix} \right] and \left[ \begin{matrix} 1 & b & c \\ b & d & e \\ c & e & f \end{matrix} \right] lead to \begin{bmatrix} -5 & -7 \\ -7 & -32 \end{bmatrix} and \begin{bmatrix} d-{ b }^{ 2 } & e-bc \\ e-bc & f-{ c }^{ 2 } \end{bmatrix}.

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