Linear Algebra Homework: Matrix Determinants & Eigenvalues

This assignment solution covers a range of linear algebra concepts, including matrix determinants, non-singularity, and trace calculations. It delves into matrix operations, such as finding cofactors, adjoints, and inverses. The solution also explores eigenvalues and eigenvectors, demonstrating how to find them and prove their properties, including orthonormality and linear independence. Furthermore, it addresses the relationship between eigenvalues and the determinant of a matrix. The assignment also touches upon the properties of idempotent matrices and their eigenvalues and includes the proof of the similarity of matrices and their eigenvalues. Finally, the solution considers linear regression models, providing a comprehensive analysis of these core concepts.

Solution: Given matrix is
a)
Note that, if A is any 3 by 3 matrix such that , then by using Laplace-
expansion by column 3, the determinant of the matrix A is
So for the given matrix,
Hence, the determinant of given matrix is 0.
b (i): we know that a matrix is non-singular if its determinant is non zero. Since the
determinant of the given matrix is 0. Hence, matrix A is not singular, that is matrix is
singular.
b (ii): The trace of A10 is the sum of diagonal elements of matrix A10.
c): Given matrix is
(i): The determinant of matrix B is
(ii): Cofactors
(iii): the cofactor matrix is
So, the adjoint of the matrix B is the transpose of matrix P. that is
(iv): The inverse of matrix B is

Solution 3:
Given matrix is
(a):
To find eigenvalues of A, solve this implies that
This implies that the eigenvalues are
(b): Since, the largest eigenvalue is . So solve we get
Now, apply gauss elimination method to find value of eigenvector u.
We get
Suppose that
This implies that
Hence, for , eigenvector corresponding to the eigenvalue is
Therefore, the eigenvector of unit length corresponding to the largest eigenvalue of A is
(c): Since, the 2nd largest eigenvalue is . So solve we get
Now, apply gauss elimination method to find value of eigenvector u.
We get
Suppose that
This implies that

Hence, for , eigenvector corresponding to the eigenvalue is
Therefore, the eigenvector of unit length corresponding to the 2nd largest eigenvalue of A
is
(d): Since and are unit vectors because norm of both vectors is 1. And the inner
product
Hence, eigenvectors are orthonormal.
(e): We know that if two vectors are orthonormal, then they are linearly independents.
From part (d), the vectors are linearly independents.
(f): We know that the determinant of the matrix is the product of eigenvalues. Since, the
eigenvalues of A are 0, 1 and 3. This implies that
. Since, determinant is 0, so the rank of matrix must be less that 3 and
hence rank of matrix is 2.
Solution 4a: To prove that if A and B are similar matrices, they have the same
eigenvalues.
Since, A and B are similar matrices, there exists a nonsingular matrix C such that C−1AC
 B.
Suppose that PA( ) and PB( ) be the characteristic polynomials of matrices A and B
respectively. This implies that
Since this implies that
… (1)
We know that if the characteristics equations of two matrices are same then their
eigenvalues are also same. From equation (1) , hence, eigenvalues of two matrices
A and B are same. This completes the proof.
Solution 4b: To prove that if A is an idempotent matrix, the eigenvalues of A are either 0
or 1.
Suppose that be an eigenvalues of idempotent matrix A and x be the corresponding
non zero Eigen vector. This implies that
.
Since, matrix A is idempotent matrix this implies that .
Now,
And,

Form equation (1) and equation (2) we get
Since, x is non zero,
Solution 4c: Given that
This implies that
We know that and also we know that trace is the sum of
diagonal elements of the given matrix. So,