Properties of the Transpose In Problems 39–42, either prove the properties in general using the fact that [ a, ]T = [a, ). or demonstrate the properties for general 3 × 3 matrices. 39. (A")T = A 40. (A + B)T = AT + BT 41. (kA) = kAT, for any scalar k 42. (AB)" = BTAT

Problem 0.6 Given a symmetric matrix A for A = [8], QA(x, y) = [x_y] [0]. (1) Check by a direct computation that QA(x, y) = ax² + 2bxy + cy². (2) Explain what {(x, y) = R² : Q₁(x, y) = 1} represents geometrically, [1], and [2], respectively. 0 (3) Diagonalize the matrix define 3 B=11 3 Can you find an orthogonal matrix V such that VBV-¹ becomes diagonal? (4) Sketch the set {(x, y) = R² : 3x² + 2xy + 3y² = 1}, by relating it with QB, and using the ingredients from (1)-(3).

1 Suppose A is a 4x4 non-singular matrix where det (A) = 3 What is the value of det (A-¹) ?