Linear Algebra Answers

1. a) Show that the eigenvalues of a hermitian matrix are real.

Apply Cramer's rule to solve the equation.

2X + Y + Z =4
X - Y +2Z =2
3X - 2Y - Z =0

How to obtain the eigenvalues and eigenvectors of the matrix: M=
[2 3 0
3 2 0
0 0 1]

Check whether the following system of equations has a solution. 4x+2y+8z+6z=3 2x+2y+2z+2w=1 x+3z+2w=3?

Obtain the eigenvalues and eigenvectors of the matrix:
M= [2 3 0]
[3 2 0]
[0 0 1]

Show for a square matrix, the followings are equivalent.
The columns of A are all vectors of length 1, and are all at right angles to each other.
A^T = A^−1

Check whether or not the matrix A=[1 1 1
0 - 2 2
0 - 2 - 3] is diagonalisable. If it is, find a matrix P, and a matrix D such that P^-1 AP=D. If A is not diagonalisable find AdjkA).

Solve the following equations using matrix algebra:

2x + y - z = 11

x - 2y + 2z = -2

3x - y + 3z = 5

solve the following equation using matrix algebra
2x+y-z=11
x-2y+2z=2
3x-y+3z=5

Solve the following equations using matrix algebra:

2x + y - z = 11

x - 2y + 2z = -2

3x - y + 3z = 5