Questions in Section Linear Algebra

Find the angle theta between bbv=-3i+4j and bbw=5i+12j. If necessary, round to the nearest tenth of a degree.

Consider the matrix B=[[2,0],[0,-1]]

1. Find the characteristic polynomial of B
2. Find the eigenvalues of B
3. Find the associated eigenvectors of B

Solve the following system of linear equations by inverting the coefficient matrix:

3x_1+5x_2=b_1

x_1+2x_2=b_2

Write a 4x4 matrix A=[a_(ij)], whose entries satisfy the following conditions:

1. a_(ij)=i^(j-1)
2. a_(ij)={(1 if | i-j | > 1),(-1 if | i-j | <= 1):}

Solve the following system of linear equations by Gauss-Jordan elimination method:

x_1+x_2+2x_3=8

-x_1-2x_2+3x_3=1

3x_1-7x_2+4x_3=10

Write the definition of a reduced row echelon form matrix. Write which of the following matrices are in the row-echelon and which are in the reduced row-echelon forms:

1. [[1,0,0],[0,0,0],[0,0,1]]
2. [[0,1,0],[1,0,0],[0,0,0]]
3. [[1,1,0],[0,1,0],[0,0,1]]
4. [[1,0,2],[0,1,3],[0,0,0]]

Consider the following system of linear equations:

x + y + 2z = a

x + z = b

2x + y + 3z = c

Show that for this system to be consistent, the constants must satisfy the condition: c=a+b.

Find the augmented matrix of the following system of linear equations:

2x_1+2x_3=1

3x_1-x_2+4x_3=7

6x_1+x_2-x_3 =0

Find inverse matrix A^(-1) by Gaussian elimination: A=([1,2,1],[1,3,1],[0,2,1]).

Write the following matrix system in matrix form. Find A^(-1) using Gauss-Jordan elimination. Use A^(-1) to then solve the system. All calculations should be exact (use fractions).

-5x+y+4z = 10.