Consider the matrix as given below. \(\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 7…
2011
Consider the matrix as given below.
\(\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 7 \\ 0 & 0 & 3\end{bmatrix}\)
Which one of the following options provides the CORRECT values of the eigenvalues of the matrix?
- A.
1,4,3
- B.
3,7,3
- C.
7,3,2
- D.
1,2,3
Attempted by 86 students.
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Correct answer: A
Key insight: For a triangular matrix (upper or lower), the eigenvalues are the entries on the main diagonal.
Identify the diagonal entries of the matrix: 1, 4, 3.
Alternatively, form the characteristic polynomial: det(A - λ I) = (1-λ)(4-λ)(3-λ).
Solve (1-λ)(4-λ)(3-λ) = 0 to get λ = 1, 4, 3.
Therefore the eigenvalues of the matrix are 1, 4, and 3 (the order of listing does not matter).
Alternate Method: The matrix is upper-triangular (all entries below the main diagonal are zero). For any triangular matrix the eigenvalues are exactly the diagonal entries. Here the diagonal entries are 1,4,3, so those are the eigenvalues.