(Solved): Problem 3 1. Express the following quadratic forms in matrix notation xAx. where A is a symmetric ...

1. Express the following quadratic forms in matrix notation $$x^{?}Ax$$. where $$A$$ is a symmetric natrix: (a) $$3x_1^2?x_1{x}_2+2x_1{r}_3+x_3^2$$ (b) $$x_1^2?3r_1{x}_2+6x_2^2$$ (c) $$6x_1{x}_3+x_2^2$$ 2. Show that writing a quadratic form in terns of a symmetric matrix is equivalent to replacing the original matrix $$A$$ by the symmetric matrix $$\left(A+A^{?}\right)/2$$. Now show that $$x^{1}Ax=x^1?\left(A+A^{?}\right)/2?x$$. 3. Find the standard form of the equation $$5x^2+4xy+2y^2=1$$. Find the orthonormal basis which yields this standard form. 4. Identify the graph of $$2x^2+y^2?4xy?4y=?4$$. 5. Identify the type of surface deseribed by $$3x^2+6y^2+3z^2?6xy?2yz=1$$. 6. Identify the type of surface described by $$2x^2+5y^2+5z^2+4xy?4xz?8\mathrm{v}z=10$$. 7. Express $$3x^2?8xy?12y^2?30x?64y=0$$ in standard form. 8. Identify the type of surlace described by $$2xy+2xz+2yz=1$$.