If a and b are 3 times 3 matrices with deta 4 and detb 1

Question

Question 1: Find a basis for the orthogonal complement of

Question 2: Find a vector that is orthogonal to both

Question 3: What is the dimension of the orthogonal complement of span

Question 4: Let W = span

(a) Find an orthonormal basis for W.

(b) What is the orthogonal projection of onto W.

(c) Write as the sum of a vector in W and a vector in W^⊥.

(d) Find the projection matrix P corresponding to orthogonal projection onto W.

Question 5: If A and B are 3 × 3 matrices with det(A) = 4 and det(B) = 1. What is the determinant of C = 2A^T A^-1BA?

Question 6: Let A be a n × n matrix with A^T = A^-1. What can you say about det(A)?

Question 7: Use Cramer's rule to solve the system:

x₁ + x₂ + x₃ = 4

x₁ - x₂ - x₃ = 0

x₁ + 2x₂ + 3x₃ = 9