LAHW#08

1. LAHW#08 Due ∞

2. 4.1. Determinants: Introduction • 2. • LetFor what values of the parameter β will the system have a unique solution?

3. 4.1. Determinants: Introduction • 8. • Use the determinantal criterion for noninvertibility (singularity) to find all the values of t for which the matrix is noninvertible (singular).

4. 4.1. Determinants: Introduction • 11. • Let A = Using Properties I, II, and III and Theorem 1, calculate the determinant of A.

5. 4.1. Determinants: Introduction • 12. • Let A = . Compute the determinant of A by using the row replacement operation only (no scaling or swapping).

6. 4.1. Determinants: Introduction • 31. • Let u = (u1, u2), v = (v1, v2), p = (u1, v1) and q = (u2, v2). Do the triangles △(0,u,v) and △(0,p,q) have the same area? (Verify or give a counterexample.) Draw the triangles involved here in a concrete case, such as u = (5, 1) and v = (4, 3), and compute the two areas in question.

7. 4.1. Determinants: Introduction • 34. • Explain why a 2 × 2 matrix A has this property: Det(αA) = α2Det(A) for α, a scalar. Then prove the corresponding result for n × n matrices.

8. 4.2. Determinant: Properties • 1. • Let Establish that f is a linear function of x.

9. 4.2. Determinants: Properties • 8. • Compute by cofactor expansions and by row operations.

10. 4.2. Determinant: Properties • 22. • Consider . Compute the determinant of this matrix using cofactor expansion.

11. 4.2. Determinant: Properties • 30. • Let A be an n × n matrix, where n is odd. Establish that if A is skew-symmetric (meaning AT = -A), then Det(A)=0.

12. 4.2. Determinants: Properties • 36. • Explain why, for two n × n matrices, Det(AB)=Det(BA). Also, explain why this does not imply that AB = BA.

13. 4.2. Determinants: Properties • 45. • Let A and B be n × n matrices such that Det(A)=14 and Det(B)=2. What are the numerical values of Det(AB), Det(BAT), Det(2A), Det(A-1), and Det(B2)?