Matrices

1.Find a change of variables that reduces the quadratic form 〖3x〗_1^2+5x_2^2+〖3x〗_3^2-2x_1 x_2+2x_(1) x_3-2x_2 x_3 to a sum of squares and express the quadratic form in terms of new variables 2.Reduce the quadratic form 2x_1 x_2+2x_1 x_3-2x_2 x_3 to canonical form by orthogonal reduction. Also find its nature. Reduce the quadratic form x^2+5y^2+z^2+2xy+2yz+6zx to canonical form and hence find its rank. 3.Reduce the quadratic form 2x_1^2+x_2^2+x_3^2+2x_1 x_2-2x_1 x_3-4x_2 x_3 to canonical form by an orthogonal transformation. Also find the rank, index, signature and nature of the quadratic form.