1. Consider the harmonic oscillator equation: d²y dy m +b + ky = 0 dt² dt a. Turn this second-order equation into a system of first-order equations b. Find the trace and determinant of the corresponding coefficient matrix for the system.

Find the matrix A of the quadratic form associated with the equation. 3x2 – 2xy + 3y2 + 8z2 – 42 = 0 A = Find the equation of the rotated quadratic surface in which the xy-, xz-, and yz-terms have been eliminated. (Use xp, yp, and zp as the new coordinates.)

Let A = [₁ ₂] be a 2 x 2 matrix with columns ₁ and a2, and let 6 € R² be a vector. Assuming the determinants det (A) = det[a₁ a2] = 6, det [az] -1, det [₁6] solve the system Az = b. S || I2 || = == -3

Find the determinant of the matrix det (M) = M = 3 0 2 0 0 3 0 -3 0 1 0 -1 -3 0 1 0