Determinants and Matrix Properties Tutorial

University:
The University Of Western Ontario
Course:
CALC 1000A | Calculus I
Academic year:

2020
Views:

453

Pages:

3
Author:

Kyuwon J.

Determinants and Matrix Properties tutorial Problem 1: Determinant of a Block Matrix (15 points) I B Let A = , where I is the n × n identity matrix, B is an n × n matrix, and C is an 0 C n × n matrix. Show that: det(A) = det(C) Solution: 1. Recall that the determinant of a block triangular matrix is the product of the determinants of its diagonal blocks. I B A= 0 C Since I and C are on the diagonal and 0 does not affect the determinant: det(A) = det(I) · det(C) 2. The determinant of the identity matrix I is 1: det(A) = 1 · det(C) = det(C) Therefore, det(A) = det(C). — Problem 2: Determinant and Eigenvalues (20 points)   2 1 1 Given A = 1 2 1, find: 1 1 2 1. The determinant of A. 2. The eigenvalues of A, and verify the relationship between the determinant and the eigenvalues. Solution: 1. Determinant of A: Use cofactor expansion: det(A) = 2 2 1 1 1 1 2 −1 +1 1 2 1 2 1 1 1 Compute: det(A) = 2(4 − 1) − 1(2 − 1) + 1(1 − 2) det(A) = 2(3) − 1(1) + 1(−1) = 6 − 1 − 1 = 4 2. Eigenvalues of A: Solve det(A − λI) = 0:   2−λ 1 1 2−λ 1  A − λI =  1 1 1 2−λ Expand: det(A − λI) = (2 − λ)((2 − λ)(2 − λ) − 1) − 1(1 · (2 − λ) − 1) + 1((2 − λ) − 1) Solve: det(A − λI) = (λ − 1)(λ − 1)(λ − 4) Eigenvalues: λ = 1, 1, 4. 3. Verification: The determinant is the product of eigenvalues: det(A) = 1 · 1 · 4 = 4 This matches the calculated determinant. — Problem 3: Properties of Inverses and Determinants (20 points) Let A and B be invertible n × n matrices. Prove that: det(A−1 BA) = det(B) Solution: 1. Using the property det(AB) = det(A) det(B): det(A−1 BA) = det(A−1 ) det(B) det(A) 2. Since det(A−1 ) = 1 : det(A) det(A−1 BA) = 1 · det(B) · det(A) det(A) 3. Simplify: det(A−1 BA) = det(B) Hence, det(A−1 BA) = det(B). — 2 Summary of Concepts • Block Matrices: The determinant of block matrices depends only on their diagonal components under specific conditions. • Eigenvalues and Determinants: The determinant of a matrix equals the product of its eigenvalues. • Matrix Inverses and Similarity Transformations: Determinants are invariant under similarity transformations (A−1 BA). Understanding these advanced properties is essential for solving problems in theoretical mathematics, physics, and engineering. 3