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

2020
Views:

51

Pages:

3
Author:

Kyuwon J.

Unit test tutorial Problem 1: Determinant and Inverse of a Matrix   2 4 1 Let A = 0 3 −1. 5 −2 0 1. Find the determinant of A. 2. If A is invertible, find its inverse. Solution: 1. Determinant: det(A) = 2 3 −1 0 −1 0 3 −4 +1 −2 0 5 0 5 −2 Expand: det(A) = 2(3 · 0 − (−1)(−2)) − 4(0 · 0 − (−1)(5)) + 1(0 · (−2) − 3 · 5) = 2(0 − 2) − 4(0 − (−5)) + 1(0 − 15) = 2(−2) − 4(5) + 1(−15) det(A) = −4 − 20 − 15 = −39 2. Inverse: Since det(A) ̸= 0, the matrix is invertible. The inverse is computed as: A−1 = 1 adj(A) det(A) Find the adjugate matrix by calculating cofactors for each element of A. After computation:   −3 −5 3 1  10 −1 −7 A−1 = −39 −15 −20 6 — Problem 2: Eigenvalues and Eigenvectors 4 1 Given B = : 2 3 1. Find the eigenvalues of B. 1 2. Find the eigenvectors corresponding to the eigenvalues. Solution: 1. Eigenvalues: The eigenvalues are found by solving: det(B − λI) = 0 4−λ 1 B − λI = 2 3−λ det(B − λI) = (4 − λ)(3 − λ) − (1)(2) = λ2 − 7λ + 10 − 2 = λ2 − 7λ + 8 = 0 Factorize: (λ − 1)(λ − 6) = 0 The eigenvalues are λ = 1 and λ = 6. 2. Eigenvectors: For λ = 1: 3 1 B − λI = 2 2 Solve (B − λI)x = 0: 3 1 2 2 x1 0 = x2 0 −1 Simplify to find x1 = . 3 1 For λ = 6, similarly find x2 = . 2 — Problem 3: System of Equations with Matrices Solve the following system using matrix methods: x + 2y + 3z = 9 2x + 3y + z = 13 3x + y + 2z = 10 Solution: 1. Write the system in matrix form:      1 2 3 x 9 2 3 1 y  = 13 3 1 2 z 10 Let A be the coefficient matrix, x the variable matrix, and b the constants: Ax = b 2 2. Find A−1 and compute: x = A−1 b After calculating A−1 and multiplying:   1 x = 2 3 So, x = 1, y = 2, z = 3. — Concepts Used and Their Importance 1. Determinants and Inverses: Used to check invertibility and solve systems of equations. Without these, solving linear systems or finding transformations would be impossible. 2. Eigenvalues and Eigenvectors: Crucial for understanding matrix properties like transformations, stability analysis, and diagonalization. 3. Matrix Multiplication: Necessary for operations like solving linear systems (A−1 b) and understanding transformations. 4. System of Equations: Solved using matrices for compactness and computational efficiency. 3