Eigenvalues And Eigenvectors

Step 1: Eigenvalue calculation.
We are given the matrix $%5Cbegin%7Bbmatrix%7D%204%20%26%202%20%5C%5C%202%20%26%204%20%5Cend%7Bbmatrix%7D$ and the eigenvector $%5Cbegin%7Bbmatrix%7D%201%20%5C%5C%202%20%5Cend%7Bbmatrix%7D$ . The general equation for an eigenvalue is: $A%20%5Cmathbf%7Bv%7D%20%3D%20%5Clambda%20%5Cmathbf%7Bv%7D$ Where $A$ is the matrix, $%5Cmathbf%7Bv%7D$ is the eigenvector, and $%5Clambda$ is the eigenvalue.

Step 2: Apply the matrix multiplication.
Multiplying the matrix $%5Cbegin%7Bbmatrix%7D%204%20%26%202%20%5C%5C%202%20%26%204%20%5Cend%7Bbmatrix%7D$ by the eigenvector $%5Cbegin%7Bbmatrix%7D%201%20%5C%5C%202%20%5Cend%7Bbmatrix%7D$ , we get: $%5Cbegin%7Bbmatrix%7D%204%20%26%202%20%5C%5C%202%20%26%204%20%5Cend%7Bbmatrix%7D%20%5Cbegin%7Bbmatrix%7D%201%20%5C%5C%202%20%5Cend%7Bbmatrix%7D%20%3D%20%5Cbegin%7Bbmatrix%7D%204%20%5Ctimes%201%20%2B%202%20%5Ctimes%202%20%5C%5C%202%20%5Ctimes%201%20%2B%204%20%5Ctimes%202%20%5Cend%7Bbmatrix%7D%20%3D%20%5Cbegin%7Bbmatrix%7D%208%20%5C%5C%2010%20%5Cend%7Bbmatrix%7D$

Step 3: Compare the result. \\ The result $%5Cbegin%7Bbmatrix%7D%208%20%5C%5C%2010%20%5Cend%7Bbmatrix%7D$ is a scalar multiple of the original eigenvector $%5Cbegin%7Bbmatrix%7D%201%20%5C%5C%202%20%5Cend%7Bbmatrix%7D$ . The scalar multiple is $4$ , which means the eigenvalue is $4$ .

Final Answer: $%5Cboxed%7B4%7D$

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

Eigenvalues And Eigenvectors

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

Chapter Questions
1 MCQs