Question

If a \(2 \times 2\) matrix \(A\) has eigenvalues 1 and 4 with the corresponding eigenvectors \(\begin{pmatrix} 1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\), respectively, then \(A\) is ________

• A. \(\begin{pmatrix} -4 & -8 \\ 5 & 9 \end{pmatrix}\)
• B. \(\begin{pmatrix} 9 & -8 \\ 5 & -4 \end{pmatrix}\)
• C. \(\begin{pmatrix} 2 & 2 \\ 1 & 3 \end{pmatrix}\)
• D. \(\begin{pmatrix} 3 & 2 \\ 1 & 2 \end{pmatrix}\)

Hint:

You can quickly check the options using trace and determinant properties.
The sum of the eigenvalues is $1 + 4 = 5$, and their product is $1 \times 4 = 4$.
The trace (sum of diagonal elements) of the correct matrix must be 5, and its determinant must be 4.
For option D: Trace $= 3+2 = 5$ and Det $= 3(2) - 2(1) = 4$. This confirms the answer instantly.

Answer 1

The Correct Option is D