Consider the following 2 × 2 matrix \(A\) where two elements are unknown and…
2015
Consider the following 2 × 2 matrix \(A\) where two elements are unknown and are marked by \(a\) and \(b\). The eigenvalues of this matrix are -1 and 7. What are the values of \(a\) and \(b\)?
\(\qquad A = \begin{pmatrix}1 & 4\\ b&a \end{pmatrix}\)
- A.
\(a = 6, b = 4\) - B.
\(a = 4, b = 6\) - C.
\(a = 3, b = 5\) - D.
\(a = 5, b =3\)
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Correct answer: D
Key idea: For a 2×2 matrix, the trace equals the sum of the eigenvalues and the determinant equals the product of the eigenvalues.
Compute the trace: trace(A) = 1 + a. The sum of the eigenvalues is -1 + 7 = 6, so 1 + a = 6, which gives a = 5.
Compute the determinant: det(A) = 1*a - 4*b = a - 4b. The product of the eigenvalues is (-1)*7 = -7, so a - 4b = -7. Substitute a = 5 to get 5 - 4b = -7, so 4b = 12 and b = 3.
Answer: a = 5, b = 3.