Linear Algebra Answers

Suppose 𝐴 and 𝑃 are 3 × 3 matrices and 𝑃^-1 exists. If 𝑃^-1𝐴𝑃 = ( 1 2 3

0 4 5

0 0 6 ) then find all eigen values of 𝐴² .

If a third degree polynomial has a lone x-intercept at x=a, discuss what this implies about the linear and quadratic factors of that polynomial.

Does A = {(x,y,z) ∈R3 : x + 3y + 2z = 0} build a set in R3

Consider the following system of linear algebraic equations 𝐴𝑥=𝑏:

[1 1 3 [ X = [2

5 3 1 Y 3

2 3 1] Z] -1]

Solve the system using (you can compare your solution with MATLAB)

1. Cramer’s Rule

2. Gauss elimination without pivoting

3. Gauss-Jordan elimination

4. LU factorization

Which one of the following sets of vectors is linearly independent?

{(1,1,0),(1,0,1),(−5,−8,4)}

{(1,1,0),(1,0,1),(4,0,4)}

{(1,1,0),(1,0,1),(2,1,1)}

{(1,1,0),(1,0,1),(−5,−9,4)}

Let  be an invertible matrix. Select which of the following statements must be true:

1.A² is invertible

2. Ax=B has a unique solution for any b

3. A can have more rows than columns

4.det(A) =0

At the beginning of a new semester, Andy makes plans for a successful semester. He allocates 42 hours per week for study time for the four courses he is taking: Math, English, Chemistry and History He decides to allocate half of his time to Math and English, and twice as much time to Math as to English. He decides to allocate twice as much to English as to History (a) Find a system of equations that represents this information. ( ) Solve the system to determine the number of hours allocated to each subject.

Give an example that the commutative law does not holds on the matrix multiplication.

Prove or Disprove that the commutative property holds on the matrix multiplication.