Question

Determining Eigenvalues of a matrix :
A. Solve the characteristic equation
B. Form matrix \(A-\lambda I\)
C. Compute determinant
D. Identify eigenvalues Choose the correct answer from the options given below :

• A. A, B, C, D
• B. B, C, A, D
• C. C, A, D, B
• D. D, A, C, B

Hint:

To find eigenvalues of a matrix: \[ A-\lambda I \rightarrow \det(A-\lambda I) \rightarrow \det(A-\lambda I)=0 \rightarrow \text{Solve for }\lambda \]

Answer 1

The Correct Option is B

Concept: Eigenvalues of a square matrix are obtained using the characteristic equation: \[ \det(A-\lambda I)=0 \] The standard procedure consists of:
• forming \(A-\lambda I\),
• computing its determinant,
• solving the characteristic equation,
• identifying eigenvalues. The sequence is extremely important.

Step 1: Forming the matrix \(A-\lambda I\). The first step is to construct: \[ A-\lambda I \] where: \[ I=\text{identity matrix} \] This introduces the eigenvalue parameter \(\lambda\).

Step 2: Computing the determinant. Next compute: \[ \det(A-\lambda I) \] This determinant produces a polynomial in \(\lambda\).

Step 3: Solving the characteristic equation. Set: \[ \det(A-\lambda I)=0 \] This equation is called the characteristic equation. Solving it gives possible eigenvalues.

Step 4: Identifying the eigenvalues. The roots obtained from the characteristic equation are the eigenvalues of the matrix.

Step 5: Arranging the correct order. Hence the proper sequence is: \[ B \to C \to A \to D \] Therefore: \[ \boxed{B,C,A,D} \] Hence the correct option is: \[ \boxed{(2)} \]