Eigenvalues and Eigenvectors Concepts

The video tutorial explains how to find an eigenvector for a 3x3 matrix with an eigenvalue of 2. It begins by setting up the equation involving the matrix, eigenvalue, and identity matrix. The process involves subtracting matrices, solving the system using an augmented matrix, and parameterizing the solution. The tutorial concludes by selecting a specific eigenvector from the possible solutions, emphasizing that it cannot be the zero vector.

What is the eigenvalue given in the problem for the 3x3 matrix?

• 1

• 2

• 4

• 3

What operation is performed on the identity matrix in the eigenvector equation setup?

• Multiplication

• Subtraction

• Division

• Addition

In the augmented matrix, which column lacks a pivot, indicating a free variable?

• Column three

• Column two

• Column one

• None

What is the value of x3 in the reduced echelon form of the augmented matrix?

• 1

• 0

• x2

• x1

How is the free variable x2 parameterized in the solution?

• x2 = -3t

• x2 = 0

• x2 = t

• x2 = 1

What is the expression for x1 in terms of the parameter t?

• x1 = 3t

• x1 = 0

• x1 = -3t

• x1 = t

What form do the eigenvectors take when parameterized by t?

• (t, -3t, 0)

• (0, 0, 0)

• (-3t, t, 0)

• (t, t, t)

Why can't the parameter t be zero when selecting an eigenvector?

• It would make x3 non-zero

• It would result in the zero vector

• It would make x2 zero

• It would make x1 zero

What is a valid eigenvector if t is chosen to be 1?

• (3, -1, 0)

• (0, 0, 0)

• (-3, 1, 0)

• (1, -3, 0)

What is the significance of scalar multiples in the context of eigenvectors?

• They change the eigenvalue

• They provide different eigenvectors

• They result in the zero vector

• They are not allowed