Consider the following matrix \(A = \left[\begin{array}{cc}2 & 3\\x & y…
2010
Consider the following matrix
\(A = \left[\begin{array}{cc}2 & 3\\x & y \end{array}\right]\)
If the eigenvalues of A are 4 and 8, then
- A.
x = 4, y = 10
- B.
x=5, y=8
- C.
x=-3, y=9
- D.
x= -4, y=10
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Correct answer: D
Key facts: For a 2x2 matrix, the trace equals the sum of eigenvalues and the determinant equals the product of eigenvalues.
Step 1: Use the trace. The sum of eigenvalues is 4 + 8 = 12, so trace(A) = 2 + y = 12, which gives y = 10.
Step 2: Use the determinant. The product of eigenvalues is 4 * 8 = 32. The determinant of A is 2*y - 3*x. Substitute y = 10 to get 2*10 - 3*x = 32, so 20 - 3x = 32.
Step 3: Solve for x. From 20 - 3x = 32 we get -3x = 12 and hence x = -4.
Conclusion: The values are x = -4 and y = 10, which satisfy both trace and determinant conditions.