The larger of the two eigenvalues of the matrix \(\begin{bmatrix} 4 & 5 \\ 2 &…
2015
The larger of the two eigenvalues of the matrix \(\begin{bmatrix}
4 & 5 \\
2 & 1 \\
\end{bmatrix}
\) is _______.
Attempted by 75 students.
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Correct answer: 6
Solution: Find the eigenvalues by solving the characteristic equation det(A − λI) = 0.
Form A − λI = [[4 − λ, 5], [2, 1 − λ]] and compute its determinant: (4 − λ)(1 − λ) − 5·2.
Expand the determinant: (4 − λ)(1 − λ) − 10 = 4 − 5λ + λ² − 10 = λ² − 5λ − 6.
Solve the quadratic λ² − 5λ − 6 = 0 using the quadratic formula: λ = [5 ± √(25 + 24)]/2 = [5 ± 7]/2.
Thus the eigenvalues are (5 + 7)/2 = 6 and (5 − 7)/2 = -1. The larger eigenvalue is 6.