Eigenvalues and Eigenvectors Analysis

The video tutorial explains how to find eigenvalues for given eigenvectors of a 2x2 matrix. It starts by introducing the concept of eigenvectors and eigenvalues, then demonstrates the process of calculating eigenvalues for two specific eigenvectors: (7, -3) and (12, -5). The tutorial provides step-by-step instructions and examples, emphasizing the relationship between eigenvectors and their corresponding eigenvalues. The video concludes with a summary of the key points discussed.

What are the given eigenvectors in the problem statement?

• (7, -3) and (12, -5)

• (7, 3) and (12, 5)

• (3, -7) and (5, -12)

• (3, 7) and (5, 12)

What equation must eigenvectors and eigenvalues satisfy?

• Matrix A minus vector x equals lambda times vector x

• Matrix A times vector x equals lambda plus vector x

• Matrix A plus vector x equals lambda times vector x

• Matrix A times vector x equals lambda times vector x

What is the eigenvalue corresponding to the eigenvector (7, -3)?

• -2

• 2

• 5

• -5

How is the eigenvalue for (7, -3) determined by inspection?

• By calculating the trace

• By using a determinant

• By solving a quadratic equation

• By recognizing the vector is a scalar multiple

What is the result of matrix A times the eigenvector (12, -5)?

• (5, -12)

• (12, -5)

• (25, -60)

• (60, -25)

What is the eigenvalue corresponding to the eigenvector (12, -5)?

• -2

• 5

• -5

• 2

How can you verify the eigenvalue using the vector equation?

• By using the trace of the matrix

• By calculating the determinant

• By forming an equation with corresponding elements

• By using the inverse of the matrix

What equation is formed using the first elements of the vectors for verification?

• 12 equals lambda plus 60

• 60 equals lambda plus 12

• 12 equals lambda times 60

• 60 equals lambda times 12

What is the scalar multiple for the vector (12, -5) to determine its eigenvalue?

• 5

• -2

• -5

• 2

What is the alternative method to determine eigenvalues if inspection is not possible?

• Calculate the trace of the matrix

• Use the determinant of the matrix

• Use the inverse of the matrix

• Set up and solve an equation using corresponding elements