Question

The determinant of a \(2 \times 2\) square matrix is 1.6. If the sum of the eigenvalues is \(-2.8\), the eigenvalues are

• A. \(-2\) and \(-0.8\)
• B. \(-1.8\) and \(-1\)
• C. \(-3\) and \(0.2\)
• D. \(-1.2\) and \(-1.6\)

Hint:

Set up the characteristic equation from the trace and determinant of the matrix.

Answer 1

The Correct Option is A

Since only the sum and the product of the eigenvalues are given, each option can be checked directly against both conditions instead of solving the quadratic.

1. $-2$ and $-0.8$: sum $= -2.8$ and product $= 1.6$. Both conditions match exactly.
2. $-1.8$ and $-1$: sum $= -2.8$ works, but product $= 1.8$, which does not equal 1.6.
3. $-3$ and $0.2$: sum $= -2.8$ works, but product $= -0.6$, which is negative and wrong.
4. $-1.2$ and $-1.6$: sum $= -2.8$ works, but product $= 1.92$, which does not equal 1.6.

Only the pair $-2$ and $-0.8$ satisfies both the given sum and the given determinant, so option A is correct.