Word Count : 656

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The given quadratic form is: $$ Q(x,y,z) = -x^2 + y^2 + z^2 + 4xy + 4xz $$ Step 1: Write the quadratic form in matrix form $$ Q(x,y,z) = \begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} -1 & 2 & 2 \\ 2 & 1 & 0 \\ 2 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} $$ Here, the symmetric matrix $A$ is: $$ A = \begin{bmatrix} -1 & 2 & 2 \\ 2 & 1 & 0 \\ 2 & 0 & 1 \end{bmatrix} $$ Step 2: Find the eigenvalues of $A$ The characteristic equation is $\det(A - \lambda I) = 0$. $$ \det \begin{bmatrix} -1-\lambda & 2 & 2 \\ 2 & 1-\lambda & 0 \\ 2 & 0 & 1-\lambda \end{bmatrix} = 0 $$ Expanding along the first row: $$ (-1-\lambda) \begin{vmatrix} 1-\lambda & 0 \\ 0 & 1-\lambda \end{vmatrix} - 2 \begin{vmatrix} 2 & 0 \\ 2 & 1-\lambda \end{vmatrix} + 2 \begin{vmatrix} ________ __________ _________ _____ _______ ________ ________ __________ ____ ________ __________ __________.
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