Discussion
The determinant of a 2 × 2 square matrix is 1.6. If the sum of the eigen values is −2.8, the eigen values are
Answer: A
The eigen values are \(-2\) and \(-0.8\).
For any square matrix the product of the eigen values equals the determinant and their sum equals the trace. So the two values satisfy
\[ \begin{aligned} \lambda_{1} + \lambda_{2} &= -2.8 \\ \lambda_{1}\lambda_{2} &= 1.6 \\ \lambda^{2} + 2.8\lambda + 1.6 &= 0 \\ (\lambda + 2)(\lambda + 0.8) &= 0 \end{aligned} \]
Check each option against both conditions. \(-1.8\) and \(-1\) add to \(-2.8\) but multiply to 1.8. \(-3\) and 0.2 add to \(-2.8\) but multiply to \(-0.6\). \(-1.2\) and \(-1.6\) multiply to 1.92 and add to \(-2.8\), so they also fail the product test. Only \(-2\) and \(-0.8\) satisfy both.
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