Solutions for MXB106 Linear Algebra Workbook 3, Semester 1, 2019

This document presents complete solutions to the Linear Algebra Workbook 3 assignment, covering key concepts such as determinants, characteristic polynomials, linear transformations, and eigenvalues/eigenvectors. The solutions include step-by-step explanations and detailed workings for each problem, ensuring a thorough understanding of the underlying principles. The document addresses the specific questions outlined in the assignment brief, providing a comprehensive guide for students studying linear algebra. The solutions are designed to aid in the comprehension of matrix operations, the properties of linear transformations, and the calculation of eigenvalues and eigenvectors. This resource is designed to help students understand the concepts related to the assignment.

Solution 1: Given matrix is
Let’s transform the matrix into upper triangular using elementary row operations.
, we get
We know that the determinant of upper triangular matrix is the product of the diagonal
entries. Therefore, the determinant of the required matrix is
Solution 2: Given matrix is
Now,
The characteristic polynomial is
Therefore, the characteristic polynomial is
Solution 3: Given that L be the linear transformation on R2 that reflects each point P
across the line
a): It is observe that vector is on the line this implies that
. Hence, is the eigenvector of L, where A is the matrix of the linear
transformation.
Now, for , the line is perpendicular to the line at the origin and if
, the line perpendicular to the line at the origin
From above, it is observed that in either case the vector is on the perpendicular line
So, by the reflection across the line , this vector is mapped to
This implies that
Hence, is the eigenvector of L.
b): Since, this implies that . So 1 is the eigenvalue.

Since, this implies that . So is the eigenvalue.
Solution 4: Given a linear transformation defined as
a): Matrix A that represents is
b): Eigenvalues and eigenvectors of matrix A:
Eigenvalues: Solve that is
Hence, the eigenvalues are
Eigenvectors: For , solve this implies that
This gives
Let
So,
Hence, eigenvector corresponding to eigenvalue 1 are
Now, for solve this implies that
This gives
So,
Hence, eigenvector corresponding to eigenvalue 2 is