Eigenvalues and Eigenvectors Concepts

The video tutorial explains how to find an eigenvector for a given 2x2 matrix with an eigenvalue of -3. It starts by setting up the equation involving the matrix and the identity matrix, then simplifies it. The tutorial proceeds to solve the augmented matrix, parameterizes the solution, and finally presents the eigenvector. The process involves understanding the role of free variables and ensuring the eigenvector is non-zero.

What is the eigenvalue given for the 2x2 matrix in the video?

• λ = 5

• λ = -3

• λ = 3

• λ = -5

What is the equation used to find the eigenvectors of a matrix?

• A - λI = 0

• A + λI = 0

• (A - λI)x = 0

• (A + λI)x = 0

What is the result of subtracting λ times the identity matrix from the given matrix?

• A matrix with all zeros

• A matrix with all ones

• A matrix with modified diagonal elements

• A matrix with unchanged elements

What is the first step in solving the augmented matrix?

• Multiplying rows

• Subtracting rows

• Row reduction to echelon form

• Adding rows

What does it mean if there is no pivot in a column of the augmented matrix?

• The column is a free variable

• The column is zero

• The column is dependent

• The column is a leading variable

How is the solution parameterized in terms of a free variable?

• By setting x1 = t

• By setting x2 = t

• By setting x1 = 0

• By setting x2 = 0

What condition must be met for the parameter t in the eigenvector solution?

• t cannot be zero

• t must be greater than zero

• t must be less than zero

• t must be zero

What is the form of the eigenvectors corresponding to λ = -3?

• Scalar multiples of (1, 0)

• Scalar multiples of (5, 3)

• Scalar multiples of (3, 5)

• Scalar multiples of (0, 1)

What is the specific eigenvector given in the video?

• (3, 5)

• (5, 3)

• (0, 1)

• (1, 0)

Why can't an eigenvector be the zero vector?

• Because it is not a solution

• Because it is not unique

• Because it is not a vector

• Because it doesn't satisfy the eigenvalue equation